A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces

Let $(x_n)$ be a sequence in a Banach space $X$ which does not converge in norm, and let $E$ be an isomorphically precisely norming set for $X$ such that $$\sum_n |x^*(x_{n+1}-x_n)|< \infty, \; \forall x^* \in E. \qquad (*)$$ Then there exists a subsequence of $(x_n)$ which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If a separable Banach space $Y$ is a separable isomorphically polyhedral then there exists a non norm convergent sequence $(x_n)$ which spans $Y$ and there exists an isomorphically precisely norming set $E$ for $Y$ such that $(*)$ is satisfied. As an application of this subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces we obtain a strengthening of a result of J. Elton, and an Orlicz-Pettis type result

A note on the method of minimal vectors

The methods of "minimal vectors" were introduced by Ansari and Enflo and strengthened by Pearcy, in order to prove the existence of hyperinvariant subspaces for certain operators on Hilbert space. In this note we present the method of minimal vectors for operators on super-reflexive Banach spaces and we give a new sufficient condition for the existence of hyperinvariant subspaces of certain operators on these spaces.Comment: Also available at http://www.math.sc.edu/~giorgis/research.htm

On the "Multiple of the Inclusion Plus Compact" Problem

The ``multiple of the inclusion plus compact problem'' which was posed by T.W. Gowers in 1996 and Th. Schlumprecht in 2003, asks whether for every infinite dimensional Banach space $X$ there exists a closed subspace $Y$ of $X$ and a bounded linear operator from $Y$ to $X$ which is not a compact perturbation of a multiple of the inclusion map from $Y$ to $X$. We give sufficient conditions on the spreading models of seminormalized basic sequences of a Banach space $X$ which guarantee that the ``multiple of the inclusion plus compact'' problem has an affirmative answer for $X$. Our results strengthen a previous result of the first named author, E.~Odell, Th. Schlumprecht and N. Tomczak-Jaegermann as well as a result of Th. Schlumprecht. We give an example of a Hereditarily Indecomposable Banach space where our results apply. For the proof of our main result we use an extension of E. Odell's Schreier unconditionality result for arrays

Quantum Kac's Chaos

We study the notion of quantum Kac's chaos which was implicitly introduced by Spohn and explicitly formulated by Gottlieb. We prove the analogue of a result of Sznitman which gives the equivalence of Kac's chaos to 2-chaoticity and to convergence of empirical measures. Finally we give a simple, different proof of a result of Spohn which states that chaos propagates with respect to certain Hamiltonians that define the evolution of the mean field limit for interacting quantum systems.Comment: The original arXiv submission is replaced in order to better reflect the content in the printed version in: Commun. Math. Sci. Vol. 16, No 7, (2018), 1801-182

A property of strictly singular 1-1 operators

We prove that if T is a strictly singular 1-1 operator defined on an infinite dimensional Banach space X, then for every infinite dimensional subspace Y of X there exists an infinite dimensional subspace Z of Y such that Z contains orbits of T of every finite length and the restriction of T on Z is a compact operator.Comment: See also: http://www.math.sc.edu/~giorgis/research.htm

GKSL Generators and Digraphs: Computing Invariant States

In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. In this article we define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be traceless as demanded by the GKSL Theorem). Moreover, we consider the converse construction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corresponding semigroup

Strictly singular, non-compact operators exist on the space of Gowers and Maurey

We construct a strictly singular non-compact operator on Gowers' and Maurey's space $GM$

The Banach space S is complementably minimal and subsequentially prime

We first include a result of the second author showing that the Banach space S is complementably minimal. We then show that every block sequence of the unit vector basis of S has a subsequence which spans a space isomorphic to its square. By the Pe{\l}czy\'nski decomposition method it follows that every basic sequence in S which spans a space complemented in S has a subsequence which spans a space isomorphic to S (i.e. S is a subsequentially prime space).Comment: See also: http://www.math.sc.edu/~giorgis/research.htm

The closedness of the generator of a semigroup

We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace transforms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector valued functions with respect to scalar measures

A new method for constructing invariant subspaces

The method of compatible sequences is introduced in order to produce non-trivial (closed) invariant subspaces of (bounded linear) operators. Also a topological tool is used which is new in the search of invariant subspaces: the extraction of continuous selections of lower semicontinuous set valued functions. The advantage of this method over previously known methods is that if an operator acts on a reflexive Banach space then it has a non-trivial invariant subspace if and only if there exist compatible sequences (their definition refers to a fixed operator). Using compatible sequences a result of Aronszajn-Smith is proved for reflexive Banach spaces. Also it is shown that if $X$ be a reflexive Banach space, $T \in {\mathcal L} (X)$, and $A$ is any closed ball of $X$, then either there exists $v \in A$ such that $Tv=0$, or there exists $v \in A$ such that $\bar{\text{Span}} \text{Orb}_T (Tv)$ is a non-trivial invariant subspace of $T$, or $A \subseteq \bar{\text{Span}} \{T^k x_{\ell} : \ell \in {\mathbb N}, 1 \leq k \leq \ell \}$ for every $(x_n)_n \in A^{\mathbb N}$