On Pitt's Theorem for Operators between Scalar and Vector-Valued Quasi-Banach Sequence Spaces

Defant, Andreas; Molina, Juan Antonio López; Rivera, M. J.

doi:10.1007/s006050050083

Cited by 8 publications

(10 citation statements)

References 8 publications

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“…The above sequence spaces are not rearrangement invariant. Further information about the Pitt theorem in rearrangement invariant setting can be found in [10] and [25].…”

Section: Introductionmentioning

confidence: 99%

Morrey Sequence Spaces: Pitt’s Theorem and Compact Embeddings

Haroske

Skrzypczak

2019

Constr Approx

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Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces mu,p = mu,p(Z d ), 0 < p ≤ u < ∞, which have yet been considered almost nowhere. They are defined as natural generalisations of the classical p spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterise the compactness of embeddings of related finite-dimensional spaces.

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“…The above sequence spaces are not rearrangement invariant. Further information about the Pitt theorem in rearrangement invariant setting can be found in [10] and [25].…”

Section: Introductionmentioning

confidence: 99%

Morrey Sequence Spaces: Pitt’s Theorem and Compact Embeddings

Haroske

Skrzypczak

2019

Constr Approx

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“…The proof is based upon the Theorem 2.7 (see [6]) which is an extension of well known Pitt's result on compact operators on p spaces (see [15]). To state the theorem we need some additional terminology.…”

Section: Proposition 24 Let F and E Be Order-continuous Banach Sequmentioning

confidence: 99%

“…We say that a Banach sequence lattice E satisfies an upper p-estimate, (respectively a lower p-estimate), if there exists a constant C > 0 such that for every choice of finitely many pairwise disjoint elements We refer the reader to [6] where some applications of the above theorem were given and examples of Banach sequence lattices satisfying lower or upper estimates were studied. Proof.…”

Section: Proposition 24 Let F and E Be Order-continuous Banach Sequmentioning

confidence: 99%

Compact multipliers on spaces of analytic functions

Mleczko

2009

Arch. Math.

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In the paper compact multiplier operators M λ : X → E from Banach spaces of analytic functions on the unit disk into Banach sequence lattices are studied. If H∞ → X → H2, then the characterization of compact multipliers is obtained through calculating the Hausdorff measure of noncompactness of diagonal operators between Banach sequence lattices. Furthermore, in the general case H∞ → X → H1, necessary and sufficient conditions for compactness are presented. Mathematics Subject Classification (2000). 42B15, 42B30, 46E05, 7B10.

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“…Pitt's theorem is the oldest and perhaps the most spectacular result along these lines and raises the question of finding more pairs (E, F ) of Banach spaces such that every continuous linear map from E into F is compact. Recently new pairs with this property have been found ( [4], [13]) and the problem has been extended to the class of quasi Banach spaces (see [5]), because of its importance in the theory of interpolation spaces. In this paper we study Pitt's theorem between general Lorentz sequence spaces p,q (µ) where 0 < p, q < ∞ and µ is a general measure on N. This study is a natural continuation (and the culmination in the case of Lorentz spaces p,q (µ)) of the work developed in [5].…”

Section: Introductionmentioning

confidence: 99%

“…Recently new pairs with this property have been found ( [4], [13]) and the problem has been extended to the class of quasi Banach spaces (see [5]), because of its importance in the theory of interpolation spaces. In this paper we study Pitt's theorem between general Lorentz sequence spaces p,q (µ) where 0 < p, q < ∞ and µ is a general measure on N. This study is a natural continuation (and the culmination in the case of Lorentz spaces p,q (µ)) of the work developed in [5]. Given quasi Banach spaces E and F , a continuous linear map T : E → F , (an operator in the sequel for short) is said to be compact if it sends bounded subsets of E into relatively compact subsets of F .…”

Section: Introductionmentioning

confidence: 99%