Spectrum (functional analysis)

<h2 id="spectrum-of-a-bounded-operator">Spectrum of a bounded operator</h2>
<h3>Definition</h3>
Let 
 
 
 
 T
 
 
 {\displaystyle T}
 
 be a <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operator</a> acting on a Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 over the complex scalar field 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, and 
 
 
 
 I
 
 
 {\displaystyle I}
 
 be the <a href="/facts/Identity_operator/9AtsP0LC">identity operator</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
. The spectrum of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is the set of all 
 
 
 
 λ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda \in \mathbb {C} }
 
 for which the operator 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 does not have an inverse that is a bounded linear operator.
Since 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is a linear operator, the inverse is linear if it exists; and, by the <a href="/facts/Bounded_inverse_theorem/KlMBa0og">bounded inverse theorem</a>, it is bounded. Therefore, the spectrum consists precisely of those scalars 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 for which 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is not <a href="/facts/Bijective/j4gTuTmW">bijective</a>.
The spectrum of a given operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is often denoted 
 
 
 
 σ
 (
 T
 )
 
 
 {\displaystyle \sigma (T)}
 
, and its complement, the <a href="/facts/Resolvent_set/f4ItQgNP">resolvent set</a>, is denoted 
 
 
 
 ρ
 (
 T
 )
 =
 
 C
 
 ∖
 σ
 (
 T
 )
 
 
 {\displaystyle \rho (T)=\mathbb {C} \setminus \sigma (T)}
 
. (
 
 
 
 ρ
 (
 T
 )
 
 
 {\displaystyle \rho (T)}
 
 is sometimes used to denote the spectral radius of 
 
 
 
 T
 
 
 {\displaystyle T}
 
)

<h3>Relation to eigenvalues</h3>
If 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is an eigenvalue of 
 
 
 
 T
 
 
 {\displaystyle T}
 
, then the operator 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is not one-to-one, and therefore its inverse 
 
 
 
 (
 T
 −
 λ
 I
 
 )
 
 −
 1
 
 
 
 
 {\displaystyle (T-\lambda I)^{-1}}
 
 is not defined. However, the converse statement is not true: the operator 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 may not have an inverse, even if 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
For example, consider the Hilbert space 
 
 
 
 
 ℓ
 
 2
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle \ell ^{2}(\mathbb {Z} )}
 
, that consists of all <a href="/facts/Sequence/gV2S4uqp">bi-infinite sequences</a> of real numbers

v
        =
        (
        …
        ,
        
          v
          
            −
            2
          
        
        ,
        
          v
          
            −
            1
          
        
        ,
        
          v
          
            0
          
        
        ,
        
          v
          
            1
          
        
        ,
        
          v
          
            2
          
        
        ,
        …
        )
      
    
    {\displaystyle v=(\ldots ,v_{-2},v_{-1},v_{0},v_{1},v_{2},\ldots )}

that have a finite sum of squares 
 
 
 
 
 ∑
 
 i
 =
 −
 ∞
 
 
 +
 ∞
 
 
 
 v
 
 i
 
 
 2
 
 
 
 
 {\textstyle \sum _{i=-\infty }^{+\infty }v_{i}^{2}}
 
. The <a href="/facts/Bilateral_shift/6c6lc0ks">bilateral shift</a> operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 simply displaces every element of the sequence by one position; namely if 
 
 
 
 u
 =
 T
 (
 v
 )
 
 
 {\displaystyle u=T(v)}
 
 then 
 
 
 
 
 u
 
 i
 
 
 =
 
 v
 
 i
 −
 1
 
 
 
 
 {\displaystyle u_{i}=v_{i-1}}
 
 for every integer 
 
 
 
 i
 
 
 {\displaystyle i}
 
. The eigenvalue equation 
 
 
 
 T
 (
 v
 )
 =
 λ
 v
 
 
 {\displaystyle T(v)=\lambda v}
 
 has no nonzero solution in this space, since it implies that all the values 
 
 
 
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i}}
 
 have the same absolute value (if 
 
 
 
 |
 λ
 |
 =
 1
 
 
 {\displaystyle \vert \lambda \vert =1}
 
) or are a geometric progression (if 
 
 
 
 |
 λ
 |
 ≠
 1
 
 
 {\displaystyle \vert \lambda \vert \neq 1}
 
); either way, the sum of their squares would not be finite. However, the operator 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is not invertible if 
 
 
 
 
 |
 
 λ
 
 |
 
 =
 1
 
 
 {\displaystyle |\lambda |=1}
 
. For example, the sequence 
 
 
 
 u
 
 
 {\displaystyle u}
 
 such that 
 
 
 
 
 u
 
 i
 
 
 =
 1
 
 /
 
 (
 
 |
 
 i
 
 |
 
 +
 1
 )
 
 
 {\displaystyle u_{i}=1/(|i|+1)}
 
 is in 
 
 
 
 
 ℓ
 
 2
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle \ell ^{2}(\mathbb {Z} )}
 
; but there is no sequence 
 
 
 
 v
 
 
 {\displaystyle v}
 
 in 
 
 
 
 
 ℓ
 
 2
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle \ell ^{2}(\mathbb {Z} )}
 
 such that 
 
 
 
 (
 T
 −
 I
 )
 v
 =
 u
 
 
 {\displaystyle (T-I)v=u}
 
 (that is, 
 
 
 
 
 v
 
 i
 −
 1
 
 
 =
 
 u
 
 i
 
 
 +
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i-1}=u_{i}+v_{i}}
 
 for all 
 
 
 
 i
 
 
 {\displaystyle i}
 
).

<h3>Basic properties</h3>
The spectrum of a bounded operator T is always a <a href="/facts/Closed_set/upYnRTUj">closed</a>, <a href="/facts/Bounded_set/yCldjtoy">bounded</a> subset of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>.
If the spectrum were empty, then the <a href="/facts/Resolvent_formalism/IKaKtcef">resolvent function</a>

R
        (
        λ
        )
        =
        (
        T
        −
        λ
        I
        
          )
          
            −
            1
          
        
        ,
        
        λ
        ∈
        
          C
        
        ,
      
    
    {\displaystyle R(\lambda )=(T-\lambda I)^{-1},\qquad \lambda \in \mathbb {C} ,}

would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function R is <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> on its domain. By the vector-valued version of <a href="/facts/Liouville%2527s_theorem_(complex_analysis)/I97twMBQ">Liouville's theorem</a>, this function is constant, thus everywhere zero as it is zero at infinity. This would be a contradiction.
The boundedness of the spectrum follows from the <a href="/facts/Neumann_series/E6e3gyAK">Neumann series expansion</a> in λ; the spectrum σ(T) is bounded by ||T||. A similar result shows the closedness of the spectrum.
The bound ||T|| on the spectrum can be refined somewhat. The <a href="/facts/Spectral_radius/j8T2xrLu">spectral radius</a>, r(T), of T is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum σ(T) inside of it, i.e.

r
        (
        T
        )
        =
        sup
        {
        
          |
        
        λ
        
          |
        
        :
        λ
        ∈
        σ
        (
        T
        )
        }
        .
      
    
    {\displaystyle r(T)=\sup\{|\lambda |:\lambda \in \sigma (T)\}.}

The spectral radius formula says<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> that for any element 
 
 
 
 T
 
 
 {\displaystyle T}
 
 of a <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a>,

r
        (
        T
        )
        =
        
          lim
          
            n
            →
            ∞
          
        
        
          
            ‖
            
              T
              
                n
              
            
            ‖
          
          
            1
            
              /
            
            n
          
        
        .
      
    
    {\displaystyle r(T)=\lim _{n\to \infty }\left\|T^{n}\right\|^{1/n}.}

<h2 id="spectrum-of-an-unbounded-operator">Spectrum of an unbounded operator</h2>
One can extend the definition of spectrum to <a href="/facts/Unbounded_operator/5u2njqHt">unbounded operators</a> on a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> X. These operators are no longer elements in the Banach algebra B(X).

<h3>Definition</h3>
Let X be a Banach space and 
 
 
 
 T
 :
 
 D
 (
 T
 )
 →
 X
 
 
 {\displaystyle T:\,D(T)\to X}
 
 be a <a href="/facts/Unbounded_operator/5u2njqHt">linear operator</a> defined on domain 
 
 
 
 D
 (
 T
 )
 ⊆
 X
 
 
 {\displaystyle D(T)\subseteq X}
 
.
A complex number λ is said to be in the resolvent set (also called regular set) of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 if the operator

T
        −
        λ
        I
        :
        
        D
        (
        T
        )
        →
        X
      
    
    {\displaystyle T-\lambda I:\,D(T)\to X}

has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator

S
        :
        
        X
        →
        D
        (
        T
        )
      
    
    {\displaystyle S:\,X\rightarrow D(T)}

such that

S
        (
        T
        −
        λ
        I
        )
        =
        
          I
          
            D
            (
            T
            )
          
        
        ,
        
        (
        T
        −
        λ
        I
        )
        S
        =
        
          I
          
            X
          
        
        .
      
    
    {\displaystyle S(T-\lambda I)=I_{D(T)},\,(T-\lambda I)S=I_{X}.}

A complex number λ is then in the spectrum if λ is not in the resolvent set.
For λ to be in the resolvent (i.e. not in the spectrum), just like in the bounded case, 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 must be bijective, since it must have a two-sided inverse. As before, if an inverse exists, then its linearity is immediate, but in general it may not be bounded, so this condition must be checked separately.
By the <a href="/facts/Closed_graph_theorem/CK2ra00W">closed graph theorem</a>, boundedness of 
 
 
 
 (
 T
 −
 λ
 I
 
 )
 
 −
 1
 
 
 
 
 {\displaystyle (T-\lambda I)^{-1}}
 
 does follow directly from its existence when T is <a href="/facts/Closed_operator/5u2njqHt">closed</a>. Then, just as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is not bijective. Note that the class of closed operators includes all bounded operators.

<h3>Basic properties</h3>
The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.
If the operator T is not <a href="/facts/Closed_linear_operator/l01mnNoJ">closed</a>, then 
 
 
 
 σ
 (
 T
 )
 =
 
 C
 
 
 
 {\displaystyle \sigma (T)=\mathbb {C} }
 
.
The following example indicates that non-closed operators may have empty spectra. Let 
 
 
 
 T
 
 
 {\displaystyle T}
 
 denote the differentiation operator on 
 
 
 
 
 L
 
 2
 
 
 (
 [
 0
 ,
 1
 ]
 )
 
 
 {\displaystyle L^{2}([0,1])}
 
, whose domain is defined to be the closure of 
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 (
 0
 ,
 1
 ]
 )
 
 
 {\displaystyle C_{c}^{\infty }((0,1])}
 
 with respect to the 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H^{1}}
 
-<a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> norm. This space can be characterized as all functions in 
 
 
 
 
 H
 
 1
 
 
 (
 [
 0
 ,
 1
 ]
 )
 
 
 {\displaystyle H^{1}([0,1])}
 
 that are zero at 
 
 
 
 t
 =
 0
 
 
 {\displaystyle t=0}
 
. Then, 
 
 
 
 T
 −
 z
 
 
 {\displaystyle T-z}
 
 has trivial kernel on this domain, as any 
 
 
 
 
 H
 
 1
 
 
 (
 [
 0
 ,
 1
 ]
 )
 
 
 {\displaystyle H^{1}([0,1])}
 
-function in its kernel is a constant multiple of 
 
 
 
 
 e
 
 z
 t
 
 
 
 
 {\displaystyle e^{zt}}
 
, which is zero at 
 
 
 
 t
 =
 0
 
 
 {\displaystyle t=0}
 
 if and only if it is identically zero. Therefore, the complement of the spectrum is all of 
 
 
 
 
 C
 
 .
 
 
 {\displaystyle \mathbb {C} .}

<h2 id="classification-of-points-in-the-spectrum">Classification of points in the spectrum</h2>
Further information: <a href="/facts/Decomposition_of_spectrum_(functional_analysis)/6jQ7KdsE">Decomposition of spectrum (functional analysis)</a>
A bounded operator T on a Banach space is invertible, i.e. has a bounded inverse, if and only if T is bounded below, i.e. 
 
 
 
 ‖
 T
 x
 ‖
 ≥
 c
 ‖
 x
 ‖
 ,
 
 
 {\displaystyle \|Tx\|\geq c\|x\|,}
 
 for some 
 
 
 
 c
 >
 0
 ,
 
 
 {\displaystyle c>0,}
 
 and has dense range. Accordingly, the spectrum of T can be divided into the following parts:

<ol><li>
 
 
 
 λ
 ∈
 σ
 (
 T
 )
 
 
 {\displaystyle \lambda \in \sigma (T)}
 
 if 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is not bounded below. In particular, this is the case if 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is not injective, that is, λ is an eigenvalue. The set of eigenvalues is called the point spectrum of T and denoted by σp(T). Alternatively, 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 could be one-to-one but still not bounded below. Such λ is not an eigenvalue but still an approximate eigenvalue of T (eigenvalues themselves are also approximate eigenvalues). The set of approximate eigenvalues (which includes the point spectrum) is called the approximate point spectrum of T, denoted by σap(T).</li>
<li>
 
 
 
 λ
 ∈
 σ
 (
 T
 )
 
 
 {\displaystyle \lambda \in \sigma (T)}
 
 if 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 does not have dense range. The set of such λ is called the compression spectrum of T, denoted by 
 
 
 
 
 σ
 
 
 c
 p
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathrm {cp} }(T)}
 
. If 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 does not have dense range but is injective, λ is said to be in the residual spectrum of T, denoted by 
 
 
 
 
 σ
 
 
 r
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathrm {r} }(T)}
 
.</li></ol>
Note that the approximate point spectrum and residual spectrum are not necessarily disjoint<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> (however, the point spectrum and the residual spectrum are).
The following subsections provide more details on the three parts of σ(T) sketched above.

<h3>Point spectrum</h3>
If an operator is not injective (so there is some nonzero x with T(x) = 0), then it is clearly not invertible. So if λ is an <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> of T, one necessarily has λ ∈ σ(T). The set of eigenvalues of T is also called the point spectrum of T, denoted by σp(T). Some authors refer to the closure of the point spectrum as the pure point spectrum 
 
 
 
 
 σ
 
 p
 p
 
 
 (
 T
 )
 =
 
 
 
 
 σ
 
 p
 
 
 (
 T
 )
 
 ¯
 
 
 
 
 {\displaystyle \sigma _{pp}(T)={\overline {\sigma _{p}(T)}}}
 
 while others simply consider 
 
 
 
 
 σ
 
 p
 p
 
 
 (
 T
 )
 :=
 
 σ
 
 p
 
 
 (
 T
 )
 .
 
 
 {\displaystyle \sigma _{pp}(T):=\sigma _{p}(T).}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h3>Approximate point spectrum</h3>
More generally, by the <a href="/facts/Bounded_inverse_theorem/KlMBa0og">bounded inverse theorem</a>, T is not invertible if it is not bounded below; that is, if there is no c > 0 such that ||Tx|| ≥ c||x|| for all x ∈ X. So the spectrum includes the set of approximate eigenvalues, which are those λ such that T - λI is not bounded below; equivalently, it is the set of λ for which there is a sequence of unit vectors x1, x2, ... for which

lim
 
 n
 →
 ∞
 
 
 ‖
 T
 
 x
 
 n
 
 
 −
 λ
 
 x
 
 n
 
 
 ‖
 =
 0
 
 
 {\displaystyle \lim _{n\to \infty }\|Tx_{n}-\lambda x_{n}\|=0}
 
.
The set of approximate eigenvalues is known as the approximate point spectrum, denoted by 
 
 
 
 
 σ
 
 
 a
 p
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathrm {ap} }(T)}
 
.
It is easy to see that the eigenvalues lie in the approximate point spectrum.
For example, consider the right shift R on 
 
 
 
 
 l
 
 2
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle l^{2}(\mathbb {Z} )}
 
 defined by

R
        :
        
        
          e
          
            j
          
        
        ↦
        
          e
          
            j
            +
            1
          
        
        ,
        
        j
        ∈
        
          Z
        
        ,
      
    
    {\displaystyle R:\,e_{j}\mapsto e_{j+1},\quad j\in \mathbb {Z} ,}

where 
 
 
 
 
 
 (
 
 
 
 e
 
 j
 
 
 
 
 
 )
 
 
 
 j
 ∈
 
 N
 
 
 
 
 
 {\displaystyle {\big (}e_{j}{\big )}_{j\in \mathbb {N} }}
 
 is the standard orthonormal basis in 
 
 
 
 
 l
 
 2
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle l^{2}(\mathbb {Z} )}
 
. Direct calculation shows R has no eigenvalues, but every λ with 
 
 
 
 
 |
 
 λ
 
 |
 
 =
 1
 
 
 {\displaystyle |\lambda |=1}
 
 is an approximate eigenvalue; letting xn be the vector

1
            
              n
            
          
        
        (
        …
        ,
        0
        ,
        1
        ,
        
          λ
          
            −
            1
          
        
        ,
        
          λ
          
            −
            2
          
        
        ,
        …
        ,
        
          λ
          
            1
            −
            n
          
        
        ,
        0
        ,
        …
        )
      
    
    {\displaystyle {\frac {1}{\sqrt {n}}}(\dots ,0,1,\lambda ^{-1},\lambda ^{-2},\dots ,\lambda ^{1-n},0,\dots )}

one can see that ||xn|| = 1 for all n, but

‖
        R
        
          x
          
            n
          
        
        −
        λ
        
          x
          
            n
          
        
        ‖
        =
        
          
            
              2
              n
            
          
        
        →
        0.
      
    
    {\displaystyle \|Rx_{n}-\lambda x_{n}\|={\sqrt {\frac {2}{n}}}\to 0.}

Since R is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum of R is its entire spectrum.
This conclusion is also true for a more general class of operators.
A unitary operator is <a href="/facts/Normal_operator/lwtz7tuF">normal</a>. By the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a>, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of H with an 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L^{2}}
 
 space) to a <a href="/facts/Multiplication_operator/YrvjuoVQ">multiplication operator</a>. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum.

<h3>Discrete spectrum</h3>
The <a href="/facts/Discrete_spectrum_(mathematics)/LoPNzT8X">discrete spectrum</a> is defined as the set of <a href="/facts/Normal_eigenvalue/0pF0hqJq">normal eigenvalues</a> or, equivalently, as the set of isolated points of the spectrum such that the corresponding <a href="/facts/Riesz_projector/QkMnvXFw">Riesz projector</a> is of finite rank. As such, the discrete spectrum is a strict subset of the point spectrum, i.e., 
 
 
 
 
 σ
 
 d
 
 
 (
 T
 )
 ⊂
 
 σ
 
 p
 
 
 (
 T
 )
 .
 
 
 {\displaystyle \sigma _{d}(T)\subset \sigma _{p}(T).}

<h3>Continuous spectrum</h3>
The set of all λ for which 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is injective and has dense range, but is not surjective, is called the continuous spectrum of T, denoted by 
 
 
 
 
 σ
 
 
 c
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathbb {c} }(T)}
 
. The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is,

σ
 
 
 c
 
 
 
 (
 T
 )
 =
 
 σ
 
 
 a
 p
 
 
 
 (
 T
 )
 ∖
 (
 
 σ
 
 
 r
 
 
 
 (
 T
 )
 ∪
 
 σ
 
 
 p
 
 
 
 (
 T
 )
 )
 
 
 {\displaystyle \sigma _{\mathrm {c} }(T)=\sigma _{\mathrm {ap} }(T)\setminus (\sigma _{\mathrm {r} }(T)\cup \sigma _{\mathrm {p} }(T))}
 
.
For example, 
 
 
 
 A
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle A:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 
 
 
 /
 
 j
 
 
 {\displaystyle e_{j}\mapsto e_{j}/j}
 
, 
 
 
 
 j
 ∈
 
 N
 
 
 
 {\displaystyle j\in \mathbb {N} }
 
, is injective and has a dense range, yet 
 
 
 
 
 R
 a
 n
 
 (
 A
 )
 ⊊
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle \mathrm {Ran} (A)\subsetneq l^{2}(\mathbb {N} )}
 
.
Indeed, if 
 
 
 
 x
 =
 
 ∑
 
 j
 ∈
 
 N
 
 
 
 
 c
 
 j
 
 
 
 e
 
 j
 
 
 ∈
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\textstyle x=\sum _{j\in \mathbb {N} }c_{j}e_{j}\in l^{2}(\mathbb {N} )}
 
 with 
 
 
 
 
 c
 
 j
 
 
 ∈
 
 C
 
 
 
 {\displaystyle c_{j}\in \mathbb {C} }
 
 such that 
 
 
 
 
 ∑
 
 j
 ∈
 
 N
 
 
 
 
 |
 
 
 c
 
 j
 
 
 
 
 |
 
 
 2
 
 
 <
 ∞
 
 
 {\textstyle \sum _{j\in \mathbb {N} }|c_{j}|^{2}<\infty }
 
, one does not necessarily have 
 
 
 
 
 ∑
 
 j
 ∈
 
 N
 
 
 
 
 
 |
 
 j
 
 c
 
 j
 
 
 
 |
 
 
 2
 
 
 <
 ∞
 
 
 {\textstyle \sum _{j\in \mathbb {N} }\left|jc_{j}\right|^{2}<\infty }
 
, and then 
 
 
 
 
 ∑
 
 j
 ∈
 
 N
 
 
 
 j
 
 c
 
 j
 
 
 
 e
 
 j
 
 
 ∉
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\textstyle \sum _{j\in \mathbb {N} }jc_{j}e_{j}\notin l^{2}(\mathbb {N} )}
 
.

<h3>Compression spectrum</h3>
The set of 
 
 
 
 λ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda \in \mathbb {C} }
 
 for which 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 does not have dense range is known as the compression spectrum of T and is denoted by 
 
 
 
 
 σ
 
 
 c
 p
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathrm {cp} }(T)}
 
.

<h3>Residual spectrum</h3>
The set of 
 
 
 
 λ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda \in \mathbb {C} }
 
 for which 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 is injective but does not have dense range is known as the residual spectrum of T and is denoted by 
 
 
 
 
 σ
 
 
 r
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathrm {r} }(T)}
 
:

σ
          
            
              r
            
          
        
        (
        T
        )
        =
        
          σ
          
            
              c
              p
            
          
        
        (
        T
        )
        ∖
        
          σ
          
            
              p
            
          
        
        (
        T
        )
        .
      
    
    {\displaystyle \sigma _{\mathrm {r} }(T)=\sigma _{\mathrm {cp} }(T)\setminus \sigma _{\mathrm {p} }(T).}

An operator may be injective, even bounded below, but still not invertible. The right shift on 
 
 
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle l^{2}(\mathbb {N} )}
 
, 
 
 
 
 R
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle R:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 R
 :
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 +
 1
 
 
 ,
 
 j
 ∈
 
 N
 
 
 
 {\displaystyle R:\,e_{j}\mapsto e_{j+1},\,j\in \mathbb {N} }
 
, is such an example. This shift operator is an <a href="/facts/Isometry/K5wX8ah6">isometry</a>, therefore bounded below by 1. But it is not invertible as it is not surjective (
 
 
 
 
 e
 
 1
 
 
 ∉
 
 R
 a
 n
 
 (
 R
 )
 
 
 {\displaystyle e_{1}\not \in \mathrm {Ran} (R)}
 
), and moreover 
 
 
 
 
 R
 a
 n
 
 (
 R
 )
 
 
 {\displaystyle \mathrm {Ran} (R)}
 
 is not dense in 
 
 
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle l^{2}(\mathbb {N} )}

(
 
 
 
 
 e
 
 1
 
 
 ∉
 
 
 
 
 R
 a
 n
 
 (
 R
 )
 
 ¯
 
 
 
 
 {\displaystyle e_{1}\notin {\overline {\mathrm {Ran} (R)}}}
 
).

<h3>Peripheral spectrum</h3>
The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Essential spectrum</h3>
There are five similar definitions of the <a href="/facts/Essential_spectrum/EHN47PNM">essential spectrum</a> of closed densely defined linear operator 
 
 
 
 A
 :
 
 X
 →
 X
 
 
 {\displaystyle A:\,X\to X}
 
 which satisfy

σ
          
            
              e
              s
              s
            
            ,
            1
          
        
        (
        A
        )
        ⊂
        
          σ
          
            
              e
              s
              s
            
            ,
            2
          
        
        (
        A
        )
        ⊂
        
          σ
          
            
              e
              s
              s
            
            ,
            3
          
        
        (
        A
        )
        ⊂
        
          σ
          
            
              e
              s
              s
            
            ,
            4
          
        
        (
        A
        )
        ⊂
        
          σ
          
            
              e
              s
              s
            
            ,
            5
          
        
        (
        A
        )
        ⊂
        σ
        (
        A
        )
        .
      
    
    {\displaystyle \sigma _{\mathrm {ess} ,1}(A)\subset \sigma _{\mathrm {ess} ,2}(A)\subset \sigma _{\mathrm {ess} ,3}(A)\subset \sigma _{\mathrm {ess} ,4}(A)\subset \sigma _{\mathrm {ess} ,5}(A)\subset \sigma (A).}

All these spectra 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 k
 
 
 (
 A
 )
 ,
  
 1
 ≤
 k
 ≤
 5
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,k}(A),\ 1\leq k\leq 5}
 
, coincide in the case of self-adjoint operators.

<ol><li>The essential spectrum 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,1}(A)}
 
 is defined as the set of points 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 of the spectrum such that 
 
 
 
 A
 −
 λ
 I
 
 
 {\displaystyle A-\lambda I}
 
 is not <a href="/facts/Fredholm_operator/FEa1xvxn">semi-Fredholm</a>. (The operator is semi-Fredholm if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.) Example 1: 
 
 
 
 λ
 =
 0
 ∈
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 A
 )
 
 
 {\displaystyle \lambda =0\in \sigma _{\mathrm {ess} ,1}(A)}
 
 for the operator 
 
 
 
 A
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle A:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 A
 :
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 
 
 
 /
 
 j
 ,
  
 j
 ∈
 
 N
 
 
 
 {\displaystyle A:\,e_{j}\mapsto e_{j}/j,~j\in \mathbb {N} }
 
 (because the range of this operator is not closed: the range does not include all of 
 
 
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle l^{2}(\mathbb {N} )}
 
 although its closure does).Example 2: 
 
 
 
 λ
 =
 0
 ∈
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 N
 )
 
 
 {\displaystyle \lambda =0\in \sigma _{\mathrm {ess} ,1}(N)}
 
 for 
 
 
 
 N
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle N:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 N
 :
 
 v
 ↦
 0
 
 
 {\displaystyle N:\,v\mapsto 0}
 
 for any 
 
 
 
 v
 ∈
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle v\in l^{2}(\mathbb {N} )}
 
 (because both kernel and cokernel of this operator are infinite-dimensional).</li>
<li>The essential spectrum 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 2
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,2}(A)}
 
 is defined as the set of points 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 of the spectrum such that the operator either 
 
 
 
 A
 −
 λ
 I
 
 
 {\displaystyle A-\lambda I}
 
 has infinite-dimensional kernel or has a range which is not closed. It can also be characterized in terms of Weyl's criterion: there exists a <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 (
 
 x
 
 j
 
 
 
 )
 
 j
 ∈
 
 N
 
 
 
 
 
 {\displaystyle (x_{j})_{j\in \mathbb {N} }}
 
 in the space X such that 
 
 
 
 ‖
 
 x
 
 j
 
 
 ‖
 =
 1
 
 
 {\displaystyle \Vert x_{j}\Vert =1}
 
, 
 
 
 
 
 lim
 
 j
 →
 ∞
 
 
 
 ‖
 
 (
 A
 −
 λ
 I
 )
 
 x
 
 j
 
 
 
 ‖
 
 =
 0
 ,
 
 
 {\textstyle \lim _{j\to \infty }\left\|(A-\lambda I)x_{j}\right\|=0,}
 
 and such that 
 
 
 
 (
 
 x
 
 j
 
 
 
 )
 
 j
 ∈
 
 N
 
 
 
 
 
 {\displaystyle (x_{j})_{j\in \mathbb {N} }}
 
 contains no convergent <a href="/facts/Subsequence/GdC2Qwau">subsequence</a>. Such a sequence is called a singular sequence (or a singular Weyl sequence).Example: 
 
 
 
 λ
 =
 0
 ∈
 
 σ
 
 
 e
 s
 s
 
 ,
 2
 
 
 (
 B
 )
 
 
 {\displaystyle \lambda =0\in \sigma _{\mathrm {ess} ,2}(B)}
 
 for the operator 
 
 
 
 B
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle B:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 B
 :
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 
 /
 
 2
 
 
 
 
 {\displaystyle B:\,e_{j}\mapsto e_{j/2}}
 
 if j is even and 
 
 
 
 
 e
 
 j
 
 
 ↦
 0
 
 
 {\displaystyle e_{j}\mapsto 0}
 
 when j is odd (kernel is infinite-dimensional; cokernel is zero-dimensional). Note that 
 
 
 
 λ
 =
 0
 ∉
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 B
 )
 
 
 {\displaystyle \lambda =0\not \in \sigma _{\mathrm {ess} ,1}(B)}
 
.</li>
<li>The essential spectrum 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 3
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,3}(A)}
 
 is defined as the set of points 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 of the spectrum such that 
 
 
 
 A
 −
 λ
 I
 
 
 {\displaystyle A-\lambda I}
 
 is not <a href="/facts/Fredholm_operator/FEa1xvxn">Fredholm</a>. (The operator is Fredholm if its range is closed and both its kernel and cokernel are finite-dimensional.) Example: 
 
 
 
 λ
 =
 0
 ∈
 
 σ
 
 
 e
 s
 s
 
 ,
 3
 
 
 (
 J
 )
 
 
 {\displaystyle \lambda =0\in \sigma _{\mathrm {ess} ,3}(J)}
 
 for the operator 
 
 
 
 J
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle J:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 J
 :
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 2
 j
 
 
 
 
 {\displaystyle J:\,e_{j}\mapsto e_{2j}}
 
 (kernel is zero-dimensional, cokernel is infinite-dimensional). Note that 
 
 
 
 λ
 =
 0
 ∉
 
 σ
 
 
 e
 s
 s
 
 ,
 2
 
 
 (
 J
 )
 
 
 {\displaystyle \lambda =0\not \in \sigma _{\mathrm {ess} ,2}(J)}
 
.</li>
<li>The essential spectrum 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 4
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,4}(A)}
 
 is defined as the set of points 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 of the spectrum such that 
 
 
 
 A
 −
 λ
 I
 
 
 {\displaystyle A-\lambda I}
 
 is not <a href="/facts/Fredholm_operator/FEa1xvxn">Fredholm</a> of index zero. It could also be characterized as the largest part of the spectrum of A which is preserved by <a href="/facts/Compact_operator/KNkcK4mE">compact</a> perturbations. In other words, 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 4
 
 
 (
 A
 )
 =
 
 ⋂
 
 K
 ∈
 
 B
 
 0
 
 
 (
 X
 )
 
 
 σ
 (
 A
 +
 K
 )
 
 
 {\textstyle \sigma _{\mathrm {ess} ,4}(A)=\bigcap _{K\in B_{0}(X)}\sigma (A+K)}
 
; here 
 
 
 
 
 B
 
 0
 
 
 (
 X
 )
 
 
 {\displaystyle B_{0}(X)}
 
 denotes the set of all compact operators on X. Example: 
 
 
 
 λ
 =
 0
 ∈
 
 σ
 
 
 e
 s
 s
 
 ,
 4
 
 
 (
 R
 )
 
 
 {\displaystyle \lambda =0\in \sigma _{\mathrm {ess} ,4}(R)}
 
 where 
 
 
 
 R
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle R:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
 is the right shift operator, 
 
 
 
 R
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle R:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 R
 :
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 +
 1
 
 
 
 
 {\displaystyle R:\,e_{j}\mapsto e_{j+1}}
 
 for 
 
 
 
 j
 ∈
 
 N
 
 
 
 {\displaystyle j\in \mathbb {N} }
 
 (its kernel is zero, its cokernel is one-dimensional). Note that 
 
 
 
 λ
 =
 0
 ∉
 
 σ
 
 
 e
 s
 s
 
 ,
 3
 
 
 (
 R
 )
 
 
 {\displaystyle \lambda =0\not \in \sigma _{\mathrm {ess} ,3}(R)}
 
.</li>
<li>The essential spectrum 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 5
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,5}(A)}
 
 is the union of 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,1}(A)}
 
 with all components of 
 
 
 
 
 C
 
 ∖
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 A
 )
 
 
 {\displaystyle \mathbb {C} \setminus \sigma _{\mathrm {ess} ,1}(A)}
 
 that do not intersect with the resolvent set 
 
 
 
 
 C
 
 ∖
 σ
 (
 A
 )
 
 
 {\displaystyle \mathbb {C} \setminus \sigma (A)}
 
. It can also be characterized as 
 
 
 
 σ
 (
 A
 )
 ∖
 
 σ
 
 
 d
 
 
 
 (
 A
 )
 
 
 {\displaystyle \sigma (A)\setminus \sigma _{\mathrm {d} }(A)}
 
.Example: consider the operator 
 
 
 
 T
 :
 
 
 l
 
 2
 
 
 (
 
 Z
 
 )
 →
 
 l
 
 2
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle T:\,l^{2}(\mathbb {Z} )\to l^{2}(\mathbb {Z} )}
 
, 
 
 
 
 T
 :
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 −
 1
 
 
 
 
 {\displaystyle T:\,e_{j}\mapsto e_{j-1}}
 
 for 
 
 
 
 j
 ≠
 0
 
 
 {\displaystyle j\neq 0}
 
, 
 
 
 
 T
 :
 
 
 e
 
 0
 
 
 ↦
 0
 
 
 {\displaystyle T:\,e_{0}\mapsto 0}
 
. Since 
 
 
 
 ‖
 T
 ‖
 =
 1
 
 
 {\displaystyle \Vert T\Vert =1}
 
, one has 
 
 
 
 σ
 (
 T
 )
 ⊂
 
 
 
 
 D
 
 
 1
 
 
 ¯
 
 
 
 
 {\displaystyle \sigma (T)\subset {\overline {\mathbb {D} _{1}}}}
 
. For any 
 
 
 
 z
 ∈
 
 C
 
 
 
 {\displaystyle z\in \mathbb {C} }
 
 with 
 
 
 
 
 |
 
 z
 
 |
 
 =
 1
 
 
 {\displaystyle |z|=1}
 
, the range of 
 
 
 
 T
 −
 z
 I
 
 
 {\displaystyle T-zI}
 
 is dense but not closed, hence the boundary of the unit disc is in the first type of the essential spectrum: 
 
 
 
 ∂
 
 
 D
 
 
 1
 
 
 ⊂
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 T
 )
 
 
 {\displaystyle \partial \mathbb {D} _{1}\subset \sigma _{\mathrm {ess} ,1}(T)}
 
. For any 
 
 
 
 z
 ∈
 
 C
 
 
 
 {\displaystyle z\in \mathbb {C} }
 
 with 
 
 
 
 
 |
 
 z
 
 |
 
 <
 1
 
 
 {\displaystyle |z|<1}
 
, 
 
 
 
 T
 −
 z
 I
 
 
 {\displaystyle T-zI}
 
 has a closed range, one-dimensional kernel, and one-dimensional cokernel, so 
 
 
 
 z
 ∈
 σ
 (
 T
 )
 
 
 {\displaystyle z\in \sigma (T)}
 
 although 
 
 
 
 z
 ∉
 
 σ
 
 
 e
 s
 s
 
 ,
 k
 
 
 (
 T
 )
 
 
 {\displaystyle z\not \in \sigma _{\mathrm {ess} ,k}(T)}
 
 for 
 
 
 
 1
 ≤
 k
 ≤
 4
 
 
 {\displaystyle 1\leq k\leq 4}
 
; thus, 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 k
 
 
 (
 T
 )
 =
 ∂
 
 
 D
 
 
 1
 
 
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,k}(T)=\partial \mathbb {D} _{1}}
 
 for 
 
 
 
 1
 ≤
 k
 ≤
 4
 
 
 {\displaystyle 1\leq k\leq 4}
 
. There are two components of 
 
 
 
 
 C
 
 ∖
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 T
 )
 
 
 {\displaystyle \mathbb {C} \setminus \sigma _{\mathrm {ess} ,1}(T)}
 
: 
 
 
 
 {
 z
 ∈
 
 C
 
 :
 
 
 |
 
 z
 
 |
 
 >
 1
 }
 
 
 {\displaystyle \{z\in \mathbb {C} :\,|z|>1\}}
 
 and 
 
 
 
 {
 z
 ∈
 
 C
 
 :
 
 
 |
 
 z
 
 |
 
 <
 1
 }
 
 
 {\displaystyle \{z\in \mathbb {C} :\,|z|<1\}}
 
. The component 
 
 
 
 {
 
 |
 
 z
 
 |
 
 <
 1
 }
 
 
 {\displaystyle \{|z|<1\}}
 
 has no intersection with the resolvent set; by definition, 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 ,
 5
 
 
 (
 T
 )
 =
 
 σ
 
 
 e
 s
 s
 
 ,
 1
 
 
 (
 T
 )
 ∪
 {
 z
 ∈
 
 C
 
 :
 
 
 |
 
 z
 
 |
 
 <
 1
 }
 =
 {
 z
 ∈
 
 C
 
 :
 
 
 |
 
 z
 
 |
 
 ≤
 1
 }
 
 
 {\displaystyle \sigma _{\mathrm {ess} ,5}(T)=\sigma _{\mathrm {ess} ,1}(T)\cup \{z\in \mathbb {C} :\,|z|<1\}=\{z\in \mathbb {C} :\,|z|\leq 1\}}
 
.</li></ol>
<h2 id="example-hydrogen-atom">Example: Hydrogen atom</h2>
The <a href="/facts/Hydrogen_atom/ncVpPbuE">hydrogen atom</a> provides an example of different types of the spectra. The <a href="/facts/Molecular_Hamiltonian/JljaHd2d">hydrogen atom Hamiltonian operator</a> 
 
 
 
 H
 =
 −
 Δ
 −
 
 
 Z
 
 
 |
 
 x
 
 |
 
 
 
 
 
 
 {\displaystyle H=-\Delta -{\frac {Z}{|x|}}}
 
, 
 
 
 
 Z
 >
 0
 
 
 {\displaystyle Z>0}
 
, with domain 
 
 
 
 D
 (
 H
 )
 =
 
 H
 
 1
 
 
 (
 
 
 R
 
 
 3
 
 
 )
 
 
 {\displaystyle D(H)=H^{1}(\mathbb {R} ^{3})}
 
 has a discrete set of eigenvalues (the discrete spectrum 
 
 
 
 
 σ
 
 
 d
 
 
 
 (
 H
 )
 
 
 {\displaystyle \sigma _{\mathrm {d} }(H)}
 
, which in this case coincides with the point spectrum 
 
 
 
 
 σ
 
 
 p
 
 
 
 (
 H
 )
 
 
 {\displaystyle \sigma _{\mathrm {p} }(H)}
 
 since there are no eigenvalues embedded into the continuous spectrum) that can be computed by the <a href="/facts/Rydberg_formula/xKRGYj5V">Rydberg formula</a>. Their corresponding <a href="/facts/Eigenfunction/62EJS0XB">eigenfunctions</a> are called eigenstates, or the <a href="/facts/Bound_state/NndW5JQl">bound states</a>. The result of the <a href="/facts/Ionization/W13Ceavd">ionization</a> process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by 
 
 
 
 
 σ
 
 
 c
 o
 n
 t
 
 
 
 (
 H
 )
 =
 [
 0
 ,
 +
 ∞
 )
 
 
 {\displaystyle \sigma _{\mathrm {cont} }(H)=[0,+\infty )}
 
 (it also coincides with the essential spectrum, 
 
 
 
 
 σ
 
 
 e
 s
 s
 
 
 
 (
 H
 )
 =
 [
 0
 ,
 +
 ∞
 )
 
 
 {\displaystyle \sigma _{\mathrm {ess} }(H)=[0,+\infty )}
 
).

<h2 id="spectrum-of-the-adjoint-operator">Spectrum of the adjoint operator</h2>
Let X be a Banach space and 
 
 
 
 T
 :
 
 X
 →
 X
 
 
 {\displaystyle T:\,X\to X}
 
 a <a href="/facts/Unbounded_operator/5u2njqHt">closed linear operator</a> with dense domain 
 
 
 
 D
 (
 T
 )
 ⊂
 X
 
 
 {\displaystyle D(T)\subset X}
 
.
If X* is the dual space of X, and 
 
 
 
 
 T
 
 ∗
 
 
 :
 
 
 X
 
 ∗
 
 
 →
 
 X
 
 ∗
 
 
 
 
 {\displaystyle T^{*}:\,X^{*}\to X^{*}}
 
 is the <a href="/facts/Hermitian_adjoint/gN6RHphd">hermitian adjoint</a> of T, then

σ
        (
        
          T
          
            ∗
          
        
        )
        =
        
          
            
              σ
              (
              T
              )
            
            ¯
          
        
        :=
        {
        z
        ∈
        
          C
        
        :
        
          
            
              z
              ¯
            
          
        
        ∈
        σ
        (
        T
        )
        }
        .
      
    
    {\displaystyle \sigma (T^{*})={\overline {\sigma (T)}}:=\{z\in \mathbb {C} :{\bar {z}}\in \sigma (T)\}.}

Theorem—For a bounded (or, more generally, closed and densely defined) operator T,

σ
 
 
 c
 p
 
 
 
 (
 T
 )
 =
 
 
 
 
 σ
 
 
 p
 
 
 
 (
 
 T
 
 ∗
 
 
 )
 
 ¯
 
 
 
 
 {\displaystyle \sigma _{\mathrm {cp} }(T)={\overline {\sigma _{\mathrm {p} }(T^{*})}}}
 
.
In particular, 
 
 
 
 
 σ
 
 
 r
 
 
 
 (
 T
 )
 ⊂
 
 
 
 
 σ
 
 
 p
 
 
 
 (
 
 T
 
 ∗
 
 
 )
 
 ¯
 
 
 ⊂
 
 σ
 
 
 r
 
 
 
 (
 T
 )
 ∪
 
 σ
 
 
 p
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathrm {r} }(T)\subset {\overline {\sigma _{\mathrm {p} }(T^{*})}}\subset \sigma _{\mathrm {r} }(T)\cup \sigma _{\mathrm {p} }(T)}
 
.

Proof
Suppose that 
 
 
 
 
 R
 a
 n
 
 (
 T
 −
 λ
 I
 )
 
 
 {\displaystyle \mathrm {Ran} (T-\lambda I)}
 
 is not dense in X. 
By the <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach theorem</a>, there exists a non-zero 
 
 
 
 φ
 ∈
 
 X
 
 ∗
 
 
 
 
 {\displaystyle \varphi \in X^{*}}
 
 that vanishes on 
 
 
 
 
 R
 a
 n
 
 (
 T
 −
 λ
 I
 )
 
 
 {\displaystyle \mathrm {Ran} (T-\lambda I)}
 
. 
For all x ∈ X,

⟨
        φ
        ,
        (
        T
        −
        λ
        I
        )
        x
        ⟩
        =
        ⟨
        (
        
          T
          
            ∗
          
        
        −
        
          
            
              λ
              ¯
            
          
        
        I
        )
        φ
        ,
        x
        ⟩
        =
        0.
      
    
    {\displaystyle \langle \varphi ,(T-\lambda I)x\rangle =\langle (T^{*}-{\bar {\lambda }}I)\varphi ,x\rangle =0.}

Therefore, 
 
 
 
 (
 
 T
 
 ∗
 
 
 −
 
 
 
 λ
 ¯
 
 
 
 I
 )
 φ
 =
 0
 ∈
 
 X
 
 ∗
 
 
 
 
 {\displaystyle (T^{*}-{\bar {\lambda }}I)\varphi =0\in X^{*}}
 
 and 
 
 
 
 
 
 
 λ
 ¯
 
 
 
 
 
 {\displaystyle {\bar {\lambda }}}
 
 is an eigenvalue of T*. 
Conversely, suppose that 
 
 
 
 
 
 
 λ
 ¯
 
 
 
 
 
 {\displaystyle {\bar {\lambda }}}
 
 is an eigenvalue of T*. Then there exists a non-zero 
 
 
 
 φ
 ∈
 
 X
 
 ∗
 
 
 
 
 {\displaystyle \varphi \in X^{*}}
 
 such that 
 
 
 
 (
 
 T
 
 ∗
 
 
 −
 
 
 
 λ
 ¯
 
 
 
 I
 )
 φ
 =
 0
 
 
 {\displaystyle (T^{*}-{\bar {\lambda }}I)\varphi =0}
 
, i.e.

∀
        x
        ∈
        X
        ,
        
        ⟨
        (
        
          T
          
            ∗
          
        
        −
        
          
            
              λ
              ¯
            
          
        
        I
        )
        φ
        ,
        x
        ⟩
        =
        ⟨
        φ
        ,
        (
        T
        −
        λ
        I
        )
        x
        ⟩
        =
        0.
      
    
    {\displaystyle \forall x\in X,\;\langle (T^{*}-{\bar {\lambda }}I)\varphi ,x\rangle =\langle \varphi ,(T-\lambda I)x\rangle =0.}

If 
 
 
 
 
 R
 a
 n
 
 (
 T
 −
 λ
 I
 )
 
 
 {\displaystyle \mathrm {Ran} (T-\lambda I)}
 
 is dense in X, then φ must be the zero functional, a contradiction. 
The claim is proved.

We also get 
 
 
 
 
 σ
 
 
 p
 
 
 
 (
 T
 )
 ⊂
 
 
 
 
 σ
 
 
 r
 
 
 
 (
 
 T
 
 ∗
 
 
 )
 ∪
 
 σ
 
 
 p
 
 
 
 (
 
 T
 
 ∗
 
 
 )
 
 ¯
 
 
 
 
 {\displaystyle \sigma _{\mathrm {p} }(T)\subset {\overline {\sigma _{\mathrm {r} }(T^{*})\cup \sigma _{\mathrm {p} }(T^{*})}}}
 
 by the following argument: X embeds isometrically into X**. 
Therefore, for every non-zero element in the kernel of 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 there exists a non-zero element in X** which vanishes on 
 
 
 
 
 R
 a
 n
 
 (
 
 T
 
 ∗
 
 
 −
 
 
 
 λ
 ¯
 
 
 
 I
 )
 
 
 {\displaystyle \mathrm {Ran} (T^{*}-{\bar {\lambda }}I)}
 
. 
Thus 
 
 
 
 
 R
 a
 n
 
 (
 
 T
 
 ∗
 
 
 −
 
 
 
 λ
 ¯
 
 
 
 I
 )
 
 
 {\displaystyle \mathrm {Ran} (T^{*}-{\bar {\lambda }}I)}
 
 can not be dense.
Furthermore, if X is reflexive, we have 
 
 
 
 
 
 
 
 σ
 
 
 r
 
 
 
 (
 
 T
 
 ∗
 
 
 )
 
 ¯
 
 
 ⊂
 
 σ
 
 
 p
 
 
 
 (
 T
 )
 
 
 {\displaystyle {\overline {\sigma _{\mathrm {r} }(T^{*})}}\subset \sigma _{\mathrm {p} }(T)}
 
.

<h2 id="spectra-of-particular-classes-of-operators">Spectra of particular classes of operators</h2>
<h3>Compact operators</h3>
If T is a <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a>, or, more generally, an <a href="/facts/Strictly_singular_operator/pgJCkK8c">inessential operator</a>, then it can be shown that the spectrum is countable, that zero is the only possible <a href="/facts/Accumulation_point/95MykoGU">accumulation point</a>, and that any nonzero λ in the spectrum is an eigenvalue.

<h3>Quasinilpotent operators</h3>
A bounded operator 
 
 
 
 A
 :
 
 X
 →
 X
 
 
 {\displaystyle A:\,X\to X}
 
 is quasinilpotent if 
 
 
 
 ‖
 
 A
 
 n
 
 
 
 ‖
 
 1
 
 /
 
 n
 
 
 →
 0
 
 
 {\displaystyle \lVert A^{n}\rVert ^{1/n}\to 0}
 
 as 
 
 
 
 n
 →
 ∞
 
 
 {\displaystyle n\to \infty }
 
 (in other words, if the spectral radius of A equals zero). Such operators could equivalently be characterized by the condition

σ
        (
        A
        )
        =
        {
        0
        }
        .
      
    
    {\displaystyle \sigma (A)=\{0\}.}

An example of such an operator is 
 
 
 
 A
 :
 
 
 l
 
 2
 
 
 (
 
 N
 
 )
 →
 
 l
 
 2
 
 
 (
 
 N
 
 )
 
 
 {\displaystyle A:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} )}
 
, 
 
 
 
 
 e
 
 j
 
 
 ↦
 
 e
 
 j
 +
 1
 
 
 
 /
 
 
 2
 
 j
 
 
 
 
 {\displaystyle e_{j}\mapsto e_{j+1}/2^{j}}
 
 for 
 
 
 
 j
 ∈
 
 N
 
 
 
 {\displaystyle j\in \mathbb {N} }
 
.

<h3>Self-adjoint operators</h3>
If X is a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> and T is a <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a> (or, more generally, a <a href="/facts/Normal_operator/lwtz7tuF">normal operator</a>), then a remarkable result known as the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
For self-adjoint operators, one can use <a href="/facts/Spectral_measure/nFN3JjlB">spectral measures</a> to define a <a href="/facts/Decomposition_of_spectrum_(functional_analysis)/6jQ7KdsE">decomposition of the spectrum</a> into absolutely continuous, pure point, and singular parts.

<h2 id="spectrum-of-a-real-operator">Spectrum of a real operator</h2>
The definitions of the resolvent and spectrum can be extended to any continuous linear operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 acting on a Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 over the real field 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 (instead of the complex field 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
) via its <a href="/facts/Complexification/fDfFLccF">complexification</a> 
 
 
 
 
 T
 
 
 C
 
 
 
 
 
 {\displaystyle T_{\mathbb {C} }}
 
. In this case we define the resolvent set 
 
 
 
 ρ
 (
 T
 )
 
 
 {\displaystyle \rho (T)}
 
 as the set of all 
 
 
 
 λ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda \in \mathbb {C} }
 
 such that 
 
 
 
 
 T
 
 
 C
 
 
 
 −
 λ
 I
 
 
 {\displaystyle T_{\mathbb {C} }-\lambda I}
 
 is invertible as an operator acting on the complexified space 
 
 
 
 
 X
 
 
 C
 
 
 
 
 
 {\displaystyle X_{\mathbb {C} }}
 
; then we define 
 
 
 
 σ
 (
 T
 )
 =
 
 C
 
 ∖
 ρ
 (
 T
 )
 
 
 {\displaystyle \sigma (T)=\mathbb {C} \setminus \rho (T)}
 
.

<h3>Real spectrum</h3>
The real spectrum of a continuous linear operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 acting on a real Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
, denoted 
 
 
 
 
 σ
 
 
 R
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma _{\mathbb {R} }(T)}
 
, is defined as the set of all 
 
 
 
 λ
 ∈
 
 R
 
 
 
 {\displaystyle \lambda \in \mathbb {R} }
 
 for which 
 
 
 
 T
 −
 λ
 I
 
 
 {\displaystyle T-\lambda I}
 
 fails to be invertible in the real algebra of bounded linear operators acting on 
 
 
 
 X
 
 
 {\displaystyle X}
 
. In this case we have 
 
 
 
 σ
 (
 T
 )
 ∩
 
 R
 
 =
 
 σ
 
 
 R
 
 
 
 (
 T
 )
 
 
 {\displaystyle \sigma (T)\cap \mathbb {R} =\sigma _{\mathbb {R} }(T)}
 
. Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.

<h2 id="spectrum-of-a-unital-banach-algebra">Spectrum of a unital Banach algebra</h2>

Let B be a complex <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a> containing a <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a> e. Then we define the spectrum σ(x) (or more explicitly σB(x)) of an element x of B to be the set of those <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> λ for which λe − x is not invertible in B. This extends the definition for bounded linear operators B(X) on a Banach space X, since B(X) is a unital Banach algebra.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Essential_spectrum/EHN47PNM">Essential spectrum</a></li>
<li><a href="/facts/Discrete_spectrum_(mathematics)/LoPNzT8X">Discrete spectrum (mathematics)</a></li>
<li><a href="/facts/Self-adjoint_operator/VrH1nEAK">Self-adjoint operator</a></li>
<li><a href="/facts/Pseudospectrum/vdAcSpPT">Pseudospectrum</a></li>
<li><a href="/facts/Resolvent_set/f4ItQgNP">Resolvent set</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Dales et al., Introduction to Banach Algebras, Operators, and Harmonic Analysis, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-53584-0</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Spectrum_of_an_operator">"Spectrum of an operator"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="/facts/Barry_Simon/BImmcE2I">Simon, Barry</a> (2005). Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3446-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2105088">2105088</a>.</li>
<li>Teschl, G. (2014). Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4704-1704-8.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kreyszig, Erwin. Introductory Functional Analysis with Applications. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Theorem 3.3.3 of Kadison & Ringrose, 1983, Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, New York: Academic Press, Inc. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Nonempty intersection between approximate point spectrum and residual spectrum". <a href="https://math.stackexchange.com/questions/1613668/nonempty-intersection-between-approximate-point-spectrum-and-residual-spectrum" target="_blank">https://math.stackexchange.com/questions/1613668/nonempty-intersection-between-approximate-point-spectrum-and-residual-spectrum</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Teschl 2014, p. 115. - Teschl, G. (2014). Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc. ISBN 978-1-4704-1704-8. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Simon 2005, p. 44. - Simon, Barry (2005). Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3446-6. MR 2105088. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2105088" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2105088</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Zaanen, Adriaan C. (2012). Introduction to Operator Theory in Riesz Spaces. Springer Science & Business Media. p. 304. ISBN 9783642606373. Retrieved 8 September 2017. <a href="9783642606373" target="_blank">9783642606373</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Spectrum (functional analysis) open-in-new

Spectrum (functional analysis)