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Erschienen in:
2018 | OriginalPaper | Buchkapitel
1. Spectrum of an Operator
verfasst von : Piotr Sołtan
Erschienen in: A Primer on Hilbert Space Operators
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Abstract
Let \(\mathcal {H}\) be a Hilbert space. The Banach space \(\mathsf {B}(\mathcal {H})\) of bounded operators on \(\mathcal {H}\) is an algebra over \(\mathbb {C}\) with multiplication defined as composition of operators and the identity operator
is the unit (neutral element of multiplication) of this algebra. The operation of passing to the adjoint operator is an anti-linear, anti-multiplicative involution
(for any \(x\in \mathsf {B}(\mathcal {H})\) we have x
∗∗ = x). Moreover, the operator norm is compatible with algebra structure in the sense that In particular \(\mathsf {B}(\mathcal {H})\) is a Banach algebra
.
is the unit (neutral element of multiplication) of this algebra. The operation of passing to the adjoint operator $$\displaystyle\mathsf {B}(\mathcal {H})\ni {x}\longmapsto {x^*}\in \mathsf {B}(\mathcal {H})$$
$$\displaystyle\|xy\|\leq \|x\|\|y\|,\qquad\quad{x,y}\in \mathsf {B}(\mathcal {H}).$$