Papers by Marek Wojtowicz
Functiones et …, 2011
The class of θ-compact spaces is introduced which properly contains the class of almost compact (... more The class of θ-compact spaces is introduced which properly contains the class of almost compact (generalized absolutely closed) spaces and is strictly contained in the class of quasicompact spaces. In the realm of almost regular spaces, the class of θ-compact spaces coincides with the class of nearly compact spaces. Moreover, an almost regular θ-compact space is mildly normal (= κ-normal). A θ-closed, θ-embedded subset of a θ-compact space is θ-compact and the product of two θ-compact space is θ-compact if one of them is compact. A (strongly) θ-continuous image of a θ-compact space is θ-compact (compact). A space is compact if and only if it is θ-compact and θ-point paracompact.
In this paper we prove the following theorem: Let (x_i) and (y_i) be unconditional bases in F-spa... more In this paper we prove the following theorem: Let (x_i) and (y_i) be unconditional bases in F-spaces X and Y, respectively, such that (x_i) is equivalent to a subbasis of (y_i) and (y_i) is equivalent to a subbasis of (x_i). Then the bases (x_i) and (y_i) are permutatively equivalent and the spaces X and Y are isomorphic.
This paper presents an elementary proof and a generalization of a theorem due to Abramovich and L... more This paper presents an elementary proof and a generalization of a theorem due to Abramovich and Lipecki, concerning the nonexistence of closed linear sublattices of finite codimension in nonatomic locally solid linear lattices with the Lebesgue property.
Journal of Mathematical Analysis and Applications
Let X be a Banach space with an unconditional basis. If X contains an isomorphic copy Y of 1 , th... more Let X be a Banach space with an unconditional basis. If X contains an isomorphic copy Y of 1 , then it contains a complemented copy of 1 located inside Y (Theorem 1). The proof is based on the possibility of constructing a projection onto a copy of 1 in X, or in a Banach function space, when the ranges of the unit vectors of 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 1 in a Banach space W and Y ⊂ W is a "slightly perturbated" copy of U , then Y is complemented in W (Lemma 2).
Bulletin of the Polish Academy of Sciences Mathematics
Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or ... more Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of \ell_1 iff for every n ≥ 1, the nth dual X^(n) of X contains no isomorphic copy of \ell_1 iff X has no quotient isomorphic to c_0 and X does not have a subspace isomorphic to \ell_1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovski˘ i, Tzafriri, Meyer-Nieberg, and Diaz and Fern´andez. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c_0, \ell_1, or a reflexive space with a Schauder basis.
Collectanea Mathematica
In the papers [2] and [4] it is proved, among other things, that every infinite dimensional $\sig... more In the papers [2] and [4] it is proved, among other things, that every infinite dimensional $\sigma$-Dedekind complete Banach lattice has a separable quotient (Corollary 2 and Theorem 2, respectively). It has come to my attention that $\textbf{L}$. Weis proved this result without assuming $\sigma$-Dedekind completeness ([3], p. 436); the proof is based, however, on the deep theorem of J. Hagler and W. B. Johnson [1] concerning the structure of dual balls of Banach spaces and therefore cannot be applied simply to the case of locally convex solid topologically complete Riesz spaces considered in ([2], Theorem 2).\newline The author wishes to thank Professor Z. Lipecki for the bibliographic information concerning the paper [5] and Proposition 1 therein leading to the above results.
Since X is isometric to a subspace of X,, (E) gives another solution to a Banach problem, posed i... more Since X is isometric to a subspace of X,, (E) gives another solution to a Banach problem, posed in [1], p. 193 (and solved for the first time in [2]): Let a Banach space X be isomorphic to a subspace of a Banach space Y and let Y be isomorphic to a subspace of X. Are then X and Y isomorphic? Let us note that this problem becomes essential (and di卤cult) if one assumes additionally that X and Y are isomorphic to complemented subspaces of each other (the Schroeder-Bernstein problem, solved in the negative by Gowers [5]). The construction in [4] is as follows: one takes Z = Z, = L1(0; 1) and its successive even duals Z,; Z,; : : :, and X is defined as ( P1 n=0Z,)`2.Of course, X, is isometric to a subspace of X. Next, the author attempts to prove that X and X, are non-isomorphic; however, Ezrohi's arguments at this point are di卤cult to follow. In the proof he refers to his Ph. D. result which, in our opinion, is incorrect. On the other hand, it is known that L1 is complemented in L,1...
Let E, G denote two Banach lattices, and let (T n ) be a sequence of continuous linear operators ... more Let E, G denote two Banach lattices, and let (T n ) be a sequence of continuous linear operators E → G. We prove that if (T n ) satises the dierence condition |T n − T m |x = |T n x − T m x| for all x ∈ E + , and if the sequence (T n x 0 ) converges for some x 0 ∈ E, then (T n ) converges pointwise on the principal ideal A x0 generated by x 0 . This result allows us to strengthen essentially an approximate-spectral theorem of the Freudenthal type obtained recently by A. W. Wickstead. MSC: 41A36, 41A65, 46B42, 47B65.
Let P denote the set of all prime numbers, and let p k denothe the kth prime. In 1873 Mertens pre... more Let P denote the set of all prime numbers, and let p k denothe the kth prime. In 1873 Mertens presented a quantitative proof of the divergence of the series P p2P
For two Banach spaces X and Y , we write dim (X) = dim (Y ) if X embeds into Y and vice versa; th... more For two Banach spaces X and Y , we write dim (X) = dim (Y ) if X embeds into Y and vice versa; then we say that X and Y have the same linear dimension. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class F has the Cantor-Bernstein property if for every X, Y ∈ F the condition dim (X) = dim (Y ) implies the respective bases (of X and Y ) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly varying Orlicz functions is of this type. This complements some results in this direction obtained earlier by S. Banach (Proposition 1.1), L. Drewnowski (Proposition 1.2), and M. J. Gonzalez, B. Sari and M. Wójtowicz (Theorem 1.4). Our theorems apply to large families of concrete Orlicz spaces.
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Papers by Marek Wojtowicz