Related papers: Operators which preserve a positive definite inner…

Let $H$ be a complex Hilbert space of dimension not less than $3$ and let ${\mathcal C}$ be a conjugacy class of compact self-adjoint operators on $H$. Suppose that the dimension of the kernels of operators from ${\mathcal C}$ not less than…

Let $A$ be a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$ and $\mathcal{B}_{A}(\mathcal{H})$ be the subspace of all operators which admit $A$-adjoints operators. In this paper, we establish some inequalities…

We show continuity properties for the pseudo-differential operator $\operatorname{Op} (a)$ from $M(\omega _0\omega ,\mathscr B )$ to $M(\omega ,\mathscr B )$, for fixed $s,\sigma \ge 1$, $\omega ,\omega _0\in \mathscr P _{s,\sigma}^0$…

Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=\int f(x)^*g(x) dx$. The assignment of the pseudodifferential…

There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture…

We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner…

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if…

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially…

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a…

Let $H$ be a complex separable Hilbert space of dimension $\geq 2$, ${\mathcal B}_s(H)$ the space of all self-adjoint operators on $H$. We give a complete classification of non-linear surjective maps on $\mathcal B_s(H)$ preserving…

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring…

It is well-known that every 6-dimensional strictly nearly K\"{a}hler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under…

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$, where…

For linear operators $L, T$ and nonlinear maps $P$, we describe classes of simple maps $F = I - P T$, $F = L - P$ between Banach and Hilbert spaces, for which no point has more than two preimages. The classes encompass known examples…

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both…

Let $X$ be a (real or complex) Banach space, and $\mathcal{I}(X)$ be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on $X$ whose squares equal themselves. We show that the Banach submanifold…

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact…

We show that if $T$ is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then $T$ is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the…

We provide sufficient and necessary conditions guaranteeing equations $(A+B)^*=A^*+B^*$ and $(AB)^*=B^*A^*$ concerning densely defined unbounded operators $A,B$ between Hilbert spaces. We also improve the perturbation theory of selfadjoint…

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni \cite{LM} of such maps for the space of compact operators on a Hilbert…