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A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 97 / Issue 3 / December 2014
- Published online by Cambridge University Press:
- 12 September 2014, pp. 331-342
- Print publication:
- December 2014
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- Article
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Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of 
$H$ is a compact operator and 
$H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that 
$H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.
