A basic sequence in a Banach space is called wide-(s) if it is bounded and dominates the summing basis. (Wide-(s) sequences were originally introduced by I. Singer, who termed them P*-sequences.) These sequences and t...
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A basic sequence in a Banach space is called wide-(s) if it is bounded and dominates the summing basis. (Wide-(s) sequences were originally introduced by I. Singer, who termed them P*-sequences.) These sequences and their quantified versions, termed lambda-wide-(s) sequences, are used to characterize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian operators, as well as a new intermediate class introduced here, the strongly Tauberian operators. This is a nonlocalizable class which nevertheless forms an open semigroup and is closed under natural operations such as taking double adjoints. It is proved for example that an operator is non-weakly compact iff for every epsilon > 0, it maps some (1 + epsilon)-wide-(s)-sequence to a wide-(s) sequence. This yields the quantitative triangular arrays result characterizing reflexivity, due to R.C. James. It is shown that an operator is non-Tauberian (resp. non-strongly Tauberian) iff for every epsilon > 0, it maps some (1 + epsilon)-wide-(s) sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than epsilon). This is applied to obtain a direct "finite" characterization of super-Tauberian operators, as well as the following characterization, which strengthens a recent result of M. Gonzalez and A. Martinez-Abejon: An operator is non-super-Tauberian iff there are for every epsilon > 0, finite (1 + epsilon)-wide-(s) sequences of arbitrary length whose images have norm at most epsilon.
Continuity of the set theoretic functions spectrum, Weyl spectrum, Browder spectrum and essential surjectivity spectrum on the classes consisting of (p, k)-quasihyponormal, M-hyponormal, totally *-paranormal and paran...
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Continuity of the set theoretic functions spectrum, Weyl spectrum, Browder spectrum and essential surjectivity spectrum on the classes consisting of (p, k)-quasihyponormal, M-hyponormal, totally *-paranormal and paranormal (Hilbert space) operators is proved.
The aim of this paper is to study L-p-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An L-p- projection on a Banach space X, for 1 <= p <= + infinity...
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The aim of this paper is to study L-p-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An L-p- projection on a Banach space X, for 1 <= p <= + infinity, is an idempotent operator P satisfying vertical bar vertical bar f vertical bar vertical bar(x) = vertical bar vertical bar(vertical bar vertical bar P(f)vertical bar vertical bar(x), vertical bar vertical bar(I-P)(f)vertical bar vertical bar(x))vertical bar vertical bar l(p) for all f is an element of X. This is an Lp version of the equality vertical bar vertical bar f vertical bar vertical bar(2) = vertical bar vertical bar Q(f)vertical bar vertical bar(2) + vertical bar vertical bar(I-Q) (f)vertical bar vertical bar(2), valid for orthogonal projections on Hilbert spaces. We study the relationships between L-p- projections on a Banach space X and those on a subspace F, as well as relationships between L-p- projections on X and those on the quotient space X/F. All the results in this paper are true for 1
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