Evgeny Semenov

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Speaker

Evgeny Mikhailovich Semenov is a plenary speaker, professor of mathematics at the Voronezh State University, Russia.

Evgeny received his Ph.D. from Voronezh University (1966) under the direction of Selin Krein and later, in 1968, his Dr. Sc. diploma also from the same university. He is the author of more than 200 papers in Functional analysis, Fourier analysis, Operator theory Approximations and expansions, Convex and discrete geometry, Global analysis. He has written also some monographs and textbooks including "Interpolation of Linear Operators" (Moscow, 1978), "Geometry of Functional Spaces" (Novosibirsk, 1979), "Нааг Series and Linear Operators" (Dordrecht: Kluwer, 1997). Нe was a speaker in many international conferences in Analysis.

Lecture

On strictly singular operators

(joint work with F.L.Hernandez, Complutense University at Madrid)

A linear operator [math]A[/math] between two Banach spaces [math]E[/math] and [math]F[/math] is called strictly singular (SS) if [math]A[/math] fails to be an isomorphism on any infinite-dimension subspace of [math]E[/math]. This concept was introduced by T.Kato. A stronger notion was introduced by B.Mityagin and A.Pelczynski. An operator [math]A[/math] from [math]E[/math] to [math]F[/math] is called super strictly singular (SSS) if the sequence of Bernstein widths [math]b_n(A)[/math] tends to [math]0[/math] when [math]n\to\infty[/math], where

[math] b_n(A)=\sup\limits_{Q\subset E, \dim Q=n} \inf\limits_{x\in Q,\|x\|_E=1} \|Ax\|_F. [/math]

In the context of Banach lattices a weaker notion was introduced by F.L.Hernandez and B.Rodriguez-Salinas. An operator [math]A[/math] from a Banach lattice [math]E[/math] to a Banach space [math]F[/math] is said to be disjointly strictly singular (DSS) if there is no disjoint sequence of non-null vectors [math](x_n)[/math] in [math]E[/math] such that the restriction of [math]A[/math] to the subspace [math][x_n][/math] is an isomorphism. It is clear that [math]K\subset SSS \subset SS\subset DSS[/math], where [math]K[/math] denotes the set of compact operators.

We present some classical and modern results about these operator sets.