Related papers: Solution of a problem of Peller concerning similar…

Given two linear operators $S$ and $T$ acting between Hilbert spaces $\mathscr{H}$ and $\mathscr{K}$, respectively $\mathscr{K}$ and $\mathscr{H}$ which satisfy the relation \begin{equation*} \langle Sh, k\rangle=\langle h, Tk\rangle, \quad…

A regular operator T on a Hilbert C^*-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I+T^*T) is dense. Semiregular operators are a slightly larger class of operators that may…

Let $H$ be a real Hilbert space. In this short note, using some of the properties of bounded linear operators with closed range defined on $H$, certain bounds for a specific convex subset of the solution set of infinite linear…

When is the collection of $\mathsf S$-Toeplitz operators with respect to a tuple of commuting bounded operators $\mathsf S= (S_1, S_2, \ldots , S_{d-1}, P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The answer is…

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results.

We consider whether L = limsup_{n to infty} n ||T^{n+1}-T^n|| < infty implies that the operator T is power bounded. We show that this is so if L<1/e, but it does not necessarily hold if L=1/e. As part of our methods, we improve a result of…

Assume that $Au=f,\quad (1)$ is a solvable linear equation in a Hilbert space $H$, $A$ is a linear, closed, densely defined, unbounded operator in $H$, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the…

We prove an analogue to the Cayley identity for an arbitrary self-adjoint operator in a Hilbert space. We also provide two new ways to characterize vectors belonging to the singular spectral subspace in terms of the analytic properties of…

The tensor complementarity problem is a specially structured nonlinear complementarity problem, then it has its particular and nice properties other than ones of the classical nonlinear complementarity problem. In this paper, it is proved…

For a closed densely defined operator $T$ from a Hilbert space $\mathfrak{H}$ to a Hilbert space $\mathfrak{K}$, necessary and sufficient conditions are established for the factorization of $T$ with a bounded nonnegative operator $X$ on…

Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator \Pi :…

For a self-adjoint unbounded operator D on a Hilbert space H, a bounded operator y on H and some complex Borel functions g(t) we establish inequalities of the type ||[g(D),y]|| \leq A|||y|| + B||[D,y]|| + ...+ X|[D, [D,...[D, y]...]]||. The…

An operator T on Hilbert space is a 3-isometry if there exists operators B and D such that (T*)^n T^n = I+nB +n^2 D. An operator J is a Jordan operator if it the sum of a unitary U and nilpotent N of order two which commute. If T is a…

We study the typical behavior of bounded linear operators on infinite dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral…

Current work defines Schmidt representation of a bilinear operator $T: H_1 \times H_2 \rightarrow K$, where $H_1, H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that…

A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. We show that if $A$ is posinormal with closed range, then $A^n$ is posinormal and has closed…

Let H be a separable Hilbert space with a fixed orthonormal basis (e_n), n>=1, and B(H) be the full von Neumann algebra of the bounded linear operators T: H -> H. Identifying l^\infty = C(\beta N) with the diagonal operators, we consider…

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a separable Hilbert space $H$ is unitarily equivalent to an operator $B$ in $H$ such that $D(A)\cap D(B)=\{0\}$. Equivalently this can be…

Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some…

For every integer $d\ge 1$, there is a unital closed subalgebra $A_d\subset B(H)$ with similarity degree equal precisely to $d$, in the sense of our previous paper. This means that for any unital homomorphism $u\colon A_d\to B(H)$ we have…