We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type,...

Probabilistic normed linear spaces (briefly PNL spaces) were first studied by A. N. Serstnev in [1]. His definition was motivated by the definition of probabilistic metric spaces (PM spaces) which were introduced by K. Menger and subsequebtly developed by A. Wald, B. Schweizer, A. Sklar and others.In a previuos paper [2] we studied the relationship between two important classes of PM spaces, namely E-spaces and pseudo-metrically generated PM spaces. We showed that a PM space is pseudo-metrically...

It is shown that the weak $L^p$ spaces ${\ell }^{p,\infty },L^{p,\infty }[0,1]$, and $L^{p,\infty }[0,\infty )$ are isomorphic as Banach spaces.

Let a compact Hausdorff space X contain a non-empty perfect subset. If α < β and β is a countable ordinal, then the Banach space Bα (X) of all bounded real-valued functions of Baire class α on X is a proper subspace of the Banach space Bβ (X). In this paper it is shown that: 1. Bα (X) has a representation as C(bα X), where bα X is a compactification of the space P X – the underlying set of X in the Baire topology generated by the Gδ -sets in X. 2. If 1 ≤ α < β ≤ Ω, where Ω is the first...

In Hörmander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate Hörmander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense...

Let $L:=-\Delta +V$ be a Schrödinger operator on ${ℝ}^n$ with $n\ge 3$ and $V\ge 0$ satisfying ${\Delta }^{-1}V\in L^{\infty }\left({ℝ}^n\right)$. Assume that $\varphi :{ℝ}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,·)$ is an Orlicz function, $\varphi (·,t)\in {𝔸}_{\infty }\left({ℝ}^n\right)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on ${ℝ}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}\left({ℝ}^n\right)\ni f\mapsto wf\in H_{\varphi }\left({ℝ}^n\right)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}\left({ℝ}^n\right)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }\left({ℝ}^n\right)$ under some assumptions on $\varphi$. As applications, the author further obtains the...

Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if AC(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.

Let ℓ be a Banach sequence space with a monotone norm $\parallel ·{\parallel }_{\ell }$, in which the canonical system $\left(e_i\right)$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_0^{\ell }\left(a\right)\times E_{\infty }^{\ell }\left(b\right)$ where $E_0^{\ell }\left(a\right)=K^{\ell }\left(exp(-p^{-1}{a}_i)\right)$ and $E_{\infty }^{\ell }\left(b\right)=K^{\ell }\left(exp\left(p{a}_i\right)\right)$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by the third author....