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Holomorphic functional calculus
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, holomorphic functional calculus is <a href="/facts/Functional_calculus/Y84ugVzS">functional calculus</a> with <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a>. That is to say, given a holomorphic function <i>f</i> of a <a href="/facts/Complex_number/2w7mqI7e">complex</a> argument <i>z</i> and an <a href="/facts/Operator_(mathematics)/1VWgEdjS">operator</a> <i>T</i>, the aim is to construct an operator, <i>f</i>(<i>T</i>), which naturally extends the function <i>f</i> from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the <a href="/facts/Spectrum_of_an_operator/cA1evTo6">spectrum</a> of <i>T</i> to the bounded operators. </p><p>This article will discuss the case where <i>T</i> is a <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operator</a> on some <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>. In particular, <i>T</i> can be a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction. </p>