arXiv subjects

Vladimir G. Troitsky

Publications and source records attributed to Vladimir G. Troitsky.

25 records · Page 2Linked to original sources

arXiv2002

Measures of non-compactness of operators on Banach lattices

Andreu et al [2] and Sadovskii [11] used representation spaces to study measures of non-compactness and spectral radii of operators on Banach lattices. In this paper, we develop representation spaces based on the nonstandard hull construction (which is equivalent to the ultrapower construction). As a particular application, we present a simple proof and some extensions of the main result of de Pagter and Schep [6] on the monotonicity of the measure of non-compactness and the spectral radius of AM-compact operators. We also use the representation spaces to characterize d-convergence and discuss the relationship between d-convergence and the measures of non-compactness.

math.FA↗

arXiv2002

A theorem of Krein revisited

M. Krein proved in 1948 that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T* has an eigenvector. We present generalizations of this result as well as some applications to C*-algebras, operators on l_1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras.

math.FA↗

arXiv2002

Spectrum of a weakly hypercyclic operator meets the unit circle

It is shown that every component of the spectrum of a weakly hypercyclic operator meets the unit circle. The proof is based on the lemma that a sequence of vectors in a Banach space whose norms grow at geometrical rate doesn't have zero in its weak closure.

math.FA↗

arXiv2001

On quasi-affine transforms of Read's operator

We show that C.J.Read's example of an operator T on l_1 which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of l_1. Furthermore, we present a bounded diagonal operator D such that even though its inverse is unbounded, but D^{-1}TD is a bounded operator with invariant subspaces, and is adjoint to an operator on c_0.

math.FA↗

arXiv2000

Spectral Radii of Bounded Operators on Topological Vector Spaces

In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several important differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern.

math.FA↗

arXiv1998

Lomonosov's theorem cannot be extended to chains of four operators

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if T is the operator on l_1 without a non-trivial closed invariant subspace constructed by C.J.Read, then there are three operators S_1, S_2 and K (non-multiples of the identity) such that T commutes with S_1, S_1 commutes with S_2, S_2 commutes with K, and K is compact. It is also shown that the commutant of T contains only series of T.

math.FA↗

arXiv1998

On the modulus of C. J. Read's operator

Let T be the quasinilpotent operator on ell_1 without an invariant subspace constructed by C. J. Read in [R3]. We prove that the modulus of this operator has an invariant subspace (and even an eigenvector). This answers a question posed by Y. Abramovich, C. Aliprantis and O. Burkinshaw in [AAB1,AAB3]

math.FA↗