Compressions of Resolvents and Maximal Radius of Regularity

Suppose that $\lambda - T$ is left-invertible in $L(H)$ for all $\lambda \in \Omega$, where $\Omega$ is an open subset of the complex plane. Then an operator-valued function $L(\lambda)$ is a left resolvent of $T$ in $\Omega$ if and only if $T$ has an extension $\tilde{T}$, the resolvent of which is a dilation of $L(\lambda)$ of a particular form. Generalized resolvents exist on every open set $U$, with $\bar{U}$ included in the regular domain of $T$. This implies a formula for the maximal radius of regularity of $T$ in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zem\'anek is obtained.Comment: 15 pages, to appear in Trans. Amer. Math. So

On mappings preserving the sharp and star orders

The present paper is devoted to the study of linear maps preserving certain relations, such as the sharp partial order and the star partial order in semisimple Banach algebras and C*-algebras.The first author is partially supported by the Spanish Ministry of Science and Innovation, D.G.I. project no. MTM2011-23843, and Junta de Andaluc´ıa grant FQM375. The second author is also supported by a Plan Propio de Investigaci´on grant from University of Almer´ıa, and Junta de Andaluc´ıa grant FQM 3737. The third author is partially supported by FEDER Funds through “Programa Operacional Factores de Competitividade – COMPETE” and by Portuguese Funds through FCT - “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, within the project PEst-OE/MAT/UI0013/2014

On the axiomatic theory of spectrum II

We give a survey of results concerning various classes of bounded linear operators in a Banach space defined by means of kernels and ranges. We show that many of these classes define a spectrum that satisfies the spectral mapping property