Extension and Lifting of Operators and Polynomials (original) (raw)

Polynomials and holomorphic functions onA-compact sets in Banach spaces

Journal of Mathematical Analysis and Applications

In this paper we study the behavior of holomorphic mappings on A-compact sets. Motivated by the recent work of Aron, Ç alişkan, García and Maestre (2016), we give several conditions (on the holomorphic mappings and on the λ-Banach operator ideal A) under which A-compact sets are preserved. Appealing to the notion of tensorstability for operator ideals, we first address the question in the polynomial setting. Then, we define a radius of (A; B)compactification that permits us to tackle the analytic case. Our approach, for instance, allows us to show that the image of any (p, r)-compact set under any holomorphic function (defined on any open set of a Banach space), is again (p, r)-compact.

Extensions of operators

Indiana University Mathematics Journal, 2004

We introduce the concept of the extension spectrum of a Hilbert space operator. This is a natural subset of the spectrum which plays an essential role in dealing with certain extension properties of operators. We prove that it has spectrallike properties and satisfies a holomorphic version of the Spectral Mapping Theorem. We establish structural theorems for algebraic extensions of triangular operators which use the extension spectrum in a natural way. The extension spectrum has some properties in common with the Kato spectrum, and in the final section we show how they are different and we examine their inclusion relationships.

On certain extension properties for the space of compact operators

1999

Let ZZZ be a fixed separable operator space, XsubsetYX\subset YXsubsetY general separable operator spaces, and T:XtoZT:X\to ZT:XtoZ a completely bounded map. ZZZ is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to YYY; the Mixed Separable Extension Property (MSEP) if every such TTT admits a bounded extension to YYY. Finally, ZZZ is said to have the Complete Separable Complementation Property (CSCP) if ZZZ is locally reflexive and TTT admits a completely bounded extension to YYY provided YYY is locally reflexive and TTT is a complete surjective isomorphism. Let bfK{\bf K}bfK denote the space of compact operators on separable Hilbert space and bfK0{\bf K}_0bfK0 the c0c_0c0 sum of CalMn{\Cal M}_nCalMn's (the space of ``small compact operators''). It is proved that bfK{\bf K}bfK has the CSCP, using the second author's previous result that bfK_0{\bf K}_0bfK0 has this property. A new proof is given for the result (due to E. Kirchberg) that bfK0{\bf K}_0bfK0 (and hence bfK{\bf K}bfK) fails the CSEP. It remains an open question if bfK{\bf K}bfK has the MSEP; it is proved this is equivalent to whether bfK0{\bf K}_0bfK_0 has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is introduced to study this problem. Further complements and open problems are discussed.

Extension of c_0(I)c_0(I)c_0(I)-valued operators on spaces of continuous functions on compact lines

2021

We investigate the problem of existence of a bounded extension to C(K) of a bounded c0(I)-valued operator T defined on the subalgebra of C(K) induced by a continuous increasing surjection φ : K → L, where K and L are compact lines. Generalizations of some of the results of [6] about extension of c0-valued operators are obtained. For instance, we prove that when a bounded extension of T exists then an extension can be obtained with norm at most twice the norm of T . Moreover, the class of compact lines L for which the c0-extension property is equivalent to the c0(I)-extension property for any continuous increasing surjection φ : K → L is studied.

Polynomially continuous operators

Israel Journal of Mathematics, 1997

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. Every compact (linear) operator is polynomially continuous. We prove that every polynomially continuous operator is weakly compact. Throughout, X and Y are Banach spaces, Sx the unit sphere of X, and N stands for the natural numbers. Given k E N, we denote by P(kX) the space of all khomogeneous (continuous) polynomials from X into the scalar field K (real or complex). We identify P(°X) = K, and denote P(X) := ~']~=o P(kX) • For the general theory of polynomials on Banach spaces, we refer to [11]. As usual, en

On analytic families of operators

Israel Journal of Mathematics, 1969

The classical Riesz-Thorin interpolation theorem [6] was extended by Hirschman [2] and Stein [5] to analytic families of operators. We recall the notions: Let F(z), z = x+iy, be analytic in 0< Re z< 1 and continuous in 0 =< Re z _< 1. F(z) is said to be of admissible growth iff Sup log iF(x + iY) I < Ae~Iyl where a < 7z. O<_x~l The significance of this notion is in the following lemma due to Hirschman [2]: LEMMA. lf F(z) is of admissible growth and ifloglF(it) l ~ ao(t), log IF(i+ it) I <= a~(t) then log I F(0) ]_<f_% Po(0, t)ao(t)dt + f 2~ P~(O, t)a~(t)dt where P~(O, t) are the values of the Poisson kernel Jor the strip, on Rez = 0, Rez = 1. We next define analytic families of linear operators: Let (M,/~) (N, v) be two measure spaces. Let {~} be a family of linear operators indexed by z, 0 ~ Re z ~ 1 so that for each z, Tz is a mapping of simple functions on M to measurable functions on N. {T~} is called an analytic family iff for any measurable set E of M of finite measure, for almost every y 6 N, the function qSr(z) = T~(X~)(y) is analytic in 0 < Re z < 1, continuous in 0 ~ Re z __< 1. The analytic family is of admissible growth iff for almost every y ~ N, ~by(z) is of admissible growth. We finally recall the notion of L(p, q) spaces. An exposition of these spaces can be found in Hunt [3]. Let f be a complex valued measurable function defined on a ~-finite measure space (M,/~). # is assumed to be non-negative. We assume that f is finite valued a.e., and denoting Ey = {x/If(x)] > Y}, 2r(y) = /~(Ey), we assume also that for some y > 0, 2;(y)< oo. We define f*(t) = Inf{y > Oily(y) < t}.

Holomorphic functions and polynomial ideals on Banach spaces

Collectanea Mathematica, 2012

Given A a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, H bA (E). We prove that, under very natural conditions satisfied by many usual classes of polynomials, the spectrum M bA (E) of this algebra "behaves" like the classical case of

Spectral theory for polynomially demicompact operators

Filomat, 2019

In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature.