We establish some versions of fixed-point theorem in a Frechet topological vector space E. The main result is that every map A BC where B is a continuous map and C is a continuous linear weakly compact operator from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Kra...

Letting E, F be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator E → F is unconditionally converging, then every polynomial from E to F is unconditionally converging (definition as in the linear case). (2) If E has the Dunford-Pettis property and every operator E → F is weakly compact, then every k-linear mapping from E into F takes wea...

In this note we show that an unbounded regular operator t on Hilbert C∗modules over an arbitrary C∗ algebra A has polar decomposition if and only if the closures of the ranges of t and |t| are orthogonally complemented, if and only if the operators t and t∗ have unbounded regular generalized inverses. For a given C∗-algebra A any densely defined A-linear closed operator t between Hilbert C∗-mod...

1.1 (Fredholm operators). Let X, Y be real Banach spaces and denote their dual spaces by X, Y . A bounded linear operator D : X → Y is called Fredholm if it has a closed image and if its kernel and cokernel (the quotient space Y/imD) are finite dimensional. Equivalently, there exists a bounded linear operator T : Y → X such that the operators TD− idX and DT − idY are compact. The Fredholm index...

We consider a closed set S ⊆ Rn and a linear operator Φ: R[X1, . . . , Xn]→ R[X1, . . . , Xn] that preserves nonnegative polynomials, in the following sense: if f ≥ 0 on S, then Φ(f) ≥ 0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which conc...

We derive some rigorous results concerning the backflow operator introduced by Bracken and Melloy. We show that it is linear bounded, self adjoint, and not compact. Thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator. From the position representation of the backflow operator we obtain a more efficient method to determine the backflow constant....

We consider the transitive linear maps on the operator algebra $B(X)$for a separable Banach space $X$. We show if a bounded linear map is norm transitive on $B(X)$,then it must be hypercyclic with strong operator topology. Also we provide a SOT-transitivelinear map without being hypercyclic in the strong operator topology.

Let X and Y be complete metric spaces of analytic functions over the unit disk in the complex plane. A linear operator T : X → Y is a bounded operator with respect to metric balls if T takes every metric ball in X into a metric ball in Y . We also say that T is metrically compact if it takes every metric ball in X into a relatively compact subset in Y . In this paper we will consider these prop...

‎let $x,y$ be normed spaces with $l(x,y)$ the space of continuous‎ ‎linear operators from $x$ into $y$‎. ‎if ${t_{j}}$ is a sequence in $l(x,y)$,‎ ‎the (bounded) multiplier space for the series $sum t_{j}$ is defined to be‎ [ ‎m^{infty}(sum t_{j})={{x_{j}}in l^{infty}(x):sum_{j=1}^{infty}%‎ ‎t_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $s:m^{infty}(sum t_{j})rightarrow y$ associat...