Revistes Catalanes amb Accés Obert (RACO)

Perturbations of the $H^\infty$-calculus

Nigel J. (Nigel John) Kalton


Suppose $A$ is a sectorial operator on a Banach space $X$, which admits an $H^\infty$-calculus. We study conditions on a multiplicative perturbation $B$ of $A$ which ensure that $B$ also has an $H^\infty$-calculus. We identify a class of bounded operators $T : X\rightarrow X$, which we call strongly triangular, such that if $B = (1+T) A$ is sectorial then it also has an $H^\infty$-calculus. In the case $X$ is a Hilbert space an operator is strongly triangular if and only if $\sum s_n(T)/n < \infty$ where $(s_n(T))^\infty_{n=1}$ are the singular values of $T$.

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