The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
Santeri Miihkinen - One of the best experts on this subject based on the ideXlab platform.
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strict singularity of a volterra type integral operator on h p
Proceedings of the American Mathematical Society, 2016Co-Authors: Santeri MiihkinenAbstract:acting on the Hardy spaces Hp of the unit disc. The operator Tg was introduced by Ch. Pommerenke and it has been studied systematically by several people including A. Aleman, A.G. Siskakis and R. Zhao among others. From a functional analytic point of view, one interesting notion is the strict singularity of a linear operator between Banach spaces. An operator is strictly singular if its restriction to any infinite-dimensional subspace is not an isomorphism onto its range. We discuss our recent result, which states that a non-compact Tg fixes an Isomorphic Copy of the sequence space lp. In particular, the strict singularity of Tg coincides with its compactness on spaces Hp.
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strict singularity of a volterra type integral operator on h p
arXiv: Functional Analysis, 2015Co-Authors: Santeri MiihkinenAbstract:We prove that a Volterra-type integral operator $T_gf(z) = \int_0^z f(\zeta)g'(\zeta)d\zeta, \, z \in \mathbb D,$ defined on Hardy spaces $H^p, \, 1 \le p < \infty,$ fixes an Isomorphic Copy of $\ell^p,$ if the operator $T_g$ is not compact. In particular, this shows that the strict singularity of the operator $T_g$ coincides with the compactness of the operator $T_g$ on spaces $H^p.$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator $T_g$ on $H^1$.
Junming Liu - One of the best experts on this subject based on the ideXlab platform.
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Strict singularity of Volterra type operators on Hardy spaces
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Qingze Lin, Junming LiuAbstract:Abstract In this paper, we first characterize the boundedness and compactness of Volterra type operator S g f ( z ) = ∫ 0 z f ′ ( ζ ) g ( ζ ) d ζ , z ∈ D , defined on Hardy spaces H p , 0 p ∞ . The spectrum of S g is also obtained. Then we prove that S g fixes an Isomorphic Copy of l p and an Isomorphic Copy of l 2 if the operator S g is not compact on H p ( 1 ≤ p ∞ ) . In particular, this implies that the strict singularity of the operator S g coincides with the compactness of the operator S g on H p . At last, we post an open question for further study.
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Strict singularity of weighted composition operators on derivative Hardy spaces
arXiv: Functional Analysis, 2019Co-Authors: Qingze Lin, Junming LiuAbstract:We prove that the weighted composition operator $W_{\phi,\varphi}$ fixes an Isomorphic Copy of $\ell^p$ if the operator $W_{\phi,\varphi}$ is not compact on the derivative Hardy space $S^p$. In particular, this implies that the strict singularity of the operator $W_{\phi,\varphi}$ coincides with the compactness of it on $S^p$. Moreover, when $p\neq2$, we characterize the conditions for those weighted composition operators $W_{\phi,\varphi}$ on $S^p$ which fix an Isomorphic Copy of $\ell^2$ .
Jon D. Vanderwerff - One of the best experts on this subject based on the ideXlab platform.
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Convex functions on Banach spaces not containing $ l_1$
Canadian Mathematical Bulletin, 1997Co-Authors: Jonathan M. Borwein, Jon D. VanderwerffAbstract:There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an Isomorphic Copy of ‡1. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.
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Convex Functions on ``Sequentially Reflexive'' Banach Spaces
1995Co-Authors: Jonathan M. Borwein, Jon D. VanderwerffAbstract:There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an Isomorphic Copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.
Miihkinen Santeri - One of the best experts on this subject based on the ideXlab platform.
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Rigidity of Volterra-type integral operators on Hardy spaces of the unit ball
2020Co-Authors: Miihkinen Santeri, Pau Jordi, Perälä Antti, Wang MaofaAbstract:We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $\mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $\ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 \le p < \infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an Isomorphic Copy of $\ell^2$ when $p \ne 2.$Comment: 18 page
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Rigidity of weighted composition operators on $H^p$
2018Co-Authors: Lindström Mikael, Miihkinen Santeri, Nieminen, Pekka J.Abstract:We show that every non-compact weighted composition operator $f \mapsto u\cdot (f\circ\phi)$ acting on a Hardy space $H^p$ for $1 \leq p < \infty$ fixes an Isomorphic Copy of the sequence space $\ell^p$ and therefore fails to be strictly singular. We also characterize those weighted composition operators on $H^p$ which fix a Copy of the Hilbert space $\ell^2$. These results extend earlier ones obtained for unweighted composition operators.Comment: 4 page
Marek Wójtowicz - One of the best experts on this subject based on the ideXlab platform.
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complemented copies of l1 in banach spaces with an unconditional basis
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Carlos Finol, Marek WójtowiczAbstract:Abstract Let X be a Banach space with an unconditional basis. If X contains an Isomorphic Copy Y of l 1 , then it contains a complemented Copy of l 1 located inside Y (Theorem 1). The proof is based on the possibility of constructing a projection onto a Copy of l 1 in X, or in a Banach function space, when the ranges of the unit vectors of l 1 are pairwise disjoint (Lemma 1). The latter result applies also to Orlicz spaces. We also show that if U is a complemented Copy of l 1 in a Banach space W and Y ⊂ W is a “slightly perturbated” Copy of U, then Y is complemented in W (Lemma 2).
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Complemented copies of ℓ1 in Banach spaces with an unconditional basis
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Carlos Finol, Marek WójtowiczAbstract:Abstract Let X be a Banach space with an unconditional basis. If X contains an Isomorphic Copy Y of l 1 , then it contains a complemented Copy of l 1 located inside Y (Theorem 1). The proof is based on the possibility of constructing a projection onto a Copy of l 1 in X, or in a Banach function space, when the ranges of the unit vectors of l 1 are pairwise disjoint (Lemma 1). The latter result applies also to Orlicz spaces. We also show that if U is a complemented Copy of l 1 in a Banach space W and Y ⊂ W is a “slightly perturbated” Copy of U, then Y is complemented in W (Lemma 2).
