Related papers: On the existence of solutions to the operator Ricc…

We consider a bounded linear operator $T$ on a complex Banach space $X$ and show that its spectral radius $r(T)$ satisfies $r(T) < 1$ if all sequences $(< x',T^nx>)_{n \in \mathbb{N}_0}$ ($x \in X$, $x' \in X'$) are, up to a certain…

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that…

A spectral analysis is done on the $L$ operator of the Lax pair for the Benjamin-Ono equation. Simplicity and finiteness of the discrete spectrum are established as are needed for the Fokas and Ablowitz inverse scattering transform scheme.…

We generalize Wonham's theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to A*P+PA-PBB*P+C*C=0 when (A,B) is exponentially stabilizable and (C,A) is exponentially…

We investigate some extremal cases of exactness constant and completely bounded projection constant. More precisely, for an $n$-dimensional operator space $E$ we prove that $\lambda_{cb}(E) = \sqrt{n}$ if and only if $ex(E) = \sqrt{n}$,…

Let A be a self-adjoint operator on a Hilbert space H. Assume that {\sigma} is an isolated component of the spectrum of A, i.e. dist({\sigma},{\Sigma})=d>0 where {\Sigma}=spec(A)\{\sigma}. Suppose that V is a bounded self-adjoint operator…

In the Celis-Dennis-Tapia (CDT) problem a quadratic function is minimized over a region defined by two strictly convex quadratic constraints. In this paper we re-derive a necessary and optimality condition for the exactness of the dual…

We study the Dirichlet problem on a bounded convex domain of $\mathbb R^N$, with zero boundary data, for truncated Laplacians ${\mathcal P}_k^\pm$, with $k<N$. We establish a necessary and sufficient condition (Theorem 1) in terms of the…

We establish sharp well-posedness and approximation estimates for variational saddle point systems at the continuous level. The main results of this note have been known to be true only in the finite dimensional case. Known spectral results…

The aim of this paper is to give strict fixed point principles for multivalued operators $T:X\rightarrow P(X)$ satisfying some contraction conditions of \'Ciri\' c and of \'Ciri\' c-Reich-Rus type. We are interested, under which conditions,…

We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$…

Let $a$ and $b$ be elements of a semisimple, complex and unital Banach algebra $A$. Using subharmonic methods, we show that if the spectral containment $\sigma(ax)\subseteq\sigma(bx)$ holds for all $x\in A$, then $ax$ belongs to the…

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert--Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is…

We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let $T, A \in B(\mathbb{X}, \mathbb{Y}),$ where $\mathbb{X}$ is a real Banach space and $\mathbb{Y}$ is a real normed linear…

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if…

One dimensional Dirac operators $$ L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi]$$, considered with $L^2$-potentials $ v(x) = 0 & P(x) Q(x) & 0$ and subject to regular boundary conditions ($bc$),…

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator $S$ obtained by restricting the self-adjoint operator $A:\D(A)\subseteq\H\to\H$ to the dense, closed with respect…

We prove existence and uniqueness of the mild solution of an infinite dimensional, operator valued, backward stochastic Riccati equation. We exploit the regularizing properties of the semigroup generated by the unbounded operator involved…

Let $\mathcal{A}$ be a $C^*$-algebra of bounded uniformly continuous functions on $X=\mathbb{R}^d$ such that $\mathcal{A}$ is stable under translations and contains the continuous functions that have a limit at infinity. Denote…

The main result of the paper is an extension of the Dirichlet problem from (closures of) bounded open domains U to arbitrary compact subsets X of the complex plane, i.e. the closure of the corresponding space of functions which are harmonic…