Compact-like operators in lattice-nonmed. spaces

Emelyanov, Eduard
Marabeh, M. A. A.
A linear operator T between two lattice-normed spaces is said to be p-compact if, for any p-bounded net x(alpha),,the net Tx(alpha) has a p-convergent subnet. p-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, AM-compact operators, etc. Similar to M-weakly and L-weakly compact operatois, we define p-M-weakly and p-L-weakly compact operators and study some of their properties. We also study up-continuous and up"compact operators between lattice nonmed vector lattices. (C) 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.


Unbounded p-Convergence in Lattice-Normed Vector Lattices
Aydın, A.; Emelyanov, Eduard; Erkurşun-Özcan, N.; Marabeh, M. (2019-07-01)
A net xα in a lattice-normed vector lattice (X, p, E) is unbounded p-convergent to x ∈ X if p(| xα− x| ∧ u) → o 0 for every u ∈ X+. This convergence has been investigated recently for (X, p, E) = (X, |·|, X) under the name of uo-convergence, for (X, p, E) = (X, ‖·‖, ℝ) under the name of un-convergence, and also for (X, p, ℝX ′) , where p(x)[f]:= |f|(|x|), under the name uaw-convergence. In this paper we study general properties of the unbounded p-convergence.
Compact-like operators in lattice-normed spaces
Aydın, Abdullah; Emel’yanov, Eduard; Department of Mathematics (2017)
Let $(X,p,E)$ and $(Y,m,F)$ be two lattice-normed spaces. A linear operator $T:Xto Y$ is said to be $p$-compact if, for any $p$-bounded net $x_alpha$ in X, the net $Tx_alpha$ has a $p$-convergent subnet in Y. That is, if $x_alpha$ is a net in X such that there is a $ein E_+$ satisfying $p(x_alpha) ≤ e$ for all $alpha$, then there exists a subnet $x_{alpha_beta}$ and $y_in Y$ such that $m(Tx_{alpha_beta} −y) xrightarrow{o}0$ in $F$. A linear operator $T:Xto Y$ is called $p$-continuous if $p(x_alpha) xrightar...
Banach lattices on which every power-bounded operator is mean ergodic
Emelyanov, Eduard (1997-01-01)
Given a Banach lattice E that fails to be countably order complete, we construct a positive compact operator A : E --> E for which T = I - A is power-bounded and not mean ergodic. As a consequence, by using the theorem of R. Zaharopol, we obtain that if every power-bounded operator in a Banach lattice is mean ergodic then the Banach lattice is reflexive.
Some finite-dimensional backward shift-invariant subspaces in the ball and a related factorization problem
Alpay, D; Kaptanoglu, HT (2000-12-15)
Beurling's theorem characterizes subspaces of the Hardy space invariant under the forward-shift operator in terms of inner functions. In this Note we consider the case where the ball replaces the open unit desk and the reproducing kernel Hilbert space with reproducing kernel 1/(1-Sigma (N)(1) a(j)w(j)*) replaces the Hardy space. We give explicit formulas which generalize Blaschke products in the case of spaces of finite codimension. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier...
Bounded operators and complemented subspaces of Cartesian products
DJAKOV, PLAMEN; TERZİOĞLU, AHMET TOSUN; Yurdakul, Murat Hayrettin; Zahariuta, V. (2011-02-01)
We study the structure of complemented subspaces in Cartesian products X x Y of Kothe spaces X and Y under the assumption that every linear continuous operator from X to Y is bounded. In particular, it is proved that each non-Montel complemented subspace with absolute basis E subset of X x Y is isomorphic to a space of the form E(1) x E(2), where E(1) is a complemented subspace of X and E(2) is a complemented subspace of Y. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Citation Formats
A. AYDIN, E. Emelyanov, N. ERKURŞUN ÖZCAN, and M. A. A. Marabeh, “Compact-like operators in lattice-nonmed. spaces,” INDAGATIONES MATHEMATICAE-NEW SERIES, pp. 633–656, 2018, Accessed: 00, 2020. [Online]. Available: