The Strong Property (B) for $$L_p$$ Spaces

Given a purely non-atomic, finite measure space $$(\Omega ,\Sigma ,\nu )$$ ( Ω , Σ , ν ) , it is proved that for every closed, infinite-dimensional subspace V of $$L_p(\nu )$$ L p ( ν ) ($$1\le p<\infty $$ 1 ≤ p < ∞ ) there exists a decomposition $$L_p(\nu )=X_1\oplus X_2$$ L p ( ν ) = X 1 ⊕ X 2 , such that both subspaces $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic to $$L_p(\nu )$$ L p ( ν ) and both $$V\cap X_1$$ V ∩ X 1 and $$V\cap X_2$$ V ∩ X 2 are infinite-dimensional. Some consequences concerning dense, non-closed range operators on $$L_1$$ L 1 are derived.

On two long standing open problems on $L_p$-spaces

The present note was written during the preparation of the talk at the International Conference dedicated to 70-th anniversary of Professor O. Lopushansky, September 16-19, 2019, Ivano-Frankivsk (Ukraine). We focus on two long standing open problems. The first one, due to Lindenstrauss and Rosenthal (1969), asks of whether every complemented infinite dimensional subspace of $L_1$ is isomorphic to either $L_1$ or $\ell_1$. The second problem was posed by Enflo and Rosenthal in 1973: does there exist a nonseparable space $L_p(\mu)$ with finite atomless $\mu$ and $1<p<\infty$, $p \neq 2$ having an unconditional basis? We analyze partial results and discuss on some natural ideas to solve these problems.

The Functional-Analytic Properties of the Limitq-Bernstein Operator

The limitq-Bernstein operatorBq,0<q<1, emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in theq-boson theory to describe the energy distribution in aq-analogue of the coherent state. Lately, the limitq-Bernstein operator has been widely under scrutiny, and it has been shown thatBqis a positive shape-preserving linear operator onC[0,1]with∥Bq∥=1. Its approximation properties, probabilistic interpretation, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties ofBqare studied. Our main result states that there exists an infinite-dimensional subspaceMofC[0,1]such that the restrictionBq|Mis an isomorphic embedding. Also we show that each such subspaceMcontains an isomorphic copy of the Banach spacec0.

The Fixed Point Property inc0with an Equivalent Norm

We study the fixed point property (FPP) in the Banach spacec0with the equivalent norm‖⋅‖D. The spacec0with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of(c0,‖⋅‖D)contains a complemented asymptotically isometric copy ofc0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of(c0,‖⋅‖D)which are notω-compact and do not contain asymptotically isometricc0—summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space(c0,‖⋅‖D),and we give some of its properties. We also prove that the dual space of(c0,‖⋅‖D)over the reals is the Bynum spacel1∞and that every infinite-dimensional subspace ofl1∞does not have the fixed point property.

An Infinite Dimensional Vector Space of Universal Functions for H∞ of the Ball

We show that there exists a closed infinite dimensional subspace of H∞(Bn) such that every function of norm one is universal for some sequence of automorphisms of Bn.

On a family of photon-added deformed coherent states associated with two-parameter deformed boson oscillator algebra

By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq < 1, Q = pq > 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q < 1 whereas for Q > 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q < 1 they show super-Poissonian behavior and there is a simultaneous squeezing in both field quadratures.PACS Nos.: 42.50.Ar, 03.65.–w

Montel Subspaces in the Countable Projective Limits of Lp(μ)-Spaces

Let us suppose one of the following conditions: (a) p ≧ 2 and F is a closed subspace of a projective limit (b) p = 1 and F is a complemented subspace of an echelon Köthe space of order 1, Λ(X,β,μ,gk); and (c) 1 > p > 2 and F is a quotient of a countable product of Lp(μn) spaces. Then, F is Montel if and only if no infinite dimensional subspace of F is normable.

Sequences and bases in p-Banach spaces

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.