Related papers: Unbounded $\sigma$-order-to-norm continuous and $u…

We consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to (mostly) positive operator measures and positive sesquilinear form measures, and investigate their relationships to…

In this manuscript, we will study both $\tilde{o}$-convergence in (partially) ordered vector spaces and a kind of convergence in a vector space $V$. A vector space $V$ is called semi-order vector space (in short semi-order space), if there…

Let $x_\alpha$ be a net in a locally solid vector lattice $(X,\tau)$; we say that $x_\alpha$ is unbounded $\tau$-convergent to a vector $x\in X$ if $\lvert x_\alpha-x \rvert\wedge w \xrightarrow{\tau} 0$ for all $w\in X_+$. In this paper,…

Recently, the functionals different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous operators with respect to unbounded convergences. We first investigate the…

In this paper, we investigate the algebras of consequence operators and finite consequence operators on a fixed language. Significant new collections of consequence operators are defined and shown to be complete and distributive…

We introduce a class of operators on $L_1$ that is stable under taking sums of pointwise unconditionally convergent series, contains all compact operators and does not contain isomorphic embeddings. It follows that any operator from $L_1$…

Given a linear semi-bounded symmetric operator $S\ge -\omega$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_\Theta$ of type $\lambda>\omega$ (i.e. generators of one-parameter…

In this paper, we introduce statistical bounded set on topological vector space. Also, we consider three classes of bounded operators from topological vector spaces to ordered topological vector spaces. Moreover, we give relations between…

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (…

We introduce a new definition of topological degree for a meaningful class of operators which need not be continuous. Subsequently, we derive a number of fixed point theorems for such operators. As an application, we deduce a new existence…

We consider the question of, given operators $A$, $Z$ and a sequence of invertible operators $U_n\to Z$, whether the sequence $U_nAU_n^{-1}$ is bounded in norm, as well as generalizations of this where $U_nAU_n^{-1}$ is modified by some…

Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a bounded linear operator. We say $T$ to be norm attaining, if there exists $x\in H_1$ with $\|x\|=1$ such that $\|Tx\|=\|T\|$. If for every closed subspace $M$ of…

The Heisenberg Oscillator Algebra admits irreducible representations both on the ring $B$ of polynomials in infinitely many indeterminates (the {\em bosonic representation}) and on a graded-by-{\em charge} vector space, the {\em…

We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately.…

Sz.-Nagy's famous theorem states that a bounded operator $T$ which acts on a complex Hilbert space $\mathcal{H}$ is similar to a unitary operator if and only if $T$ is invertible and both $T$ and $T^{-1}$ are power bounded. There is an…

The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of…

Suppose $\Sigma$ is a topological space and $S(\Sigma)$ is the vector lattice of all equivalent classes of continuous real-valued functions defined on open dense subsets of $\Sigma$. In this paper, we establish some lattice and topological…

In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit…

An operator $T$ on a Hilbert space $\mathcal H$ is called expansive, if $\|Tx\|\geq \|x\|$ ($x\in\mathcal H$). Expansive operators $T$ quasisimilar to the unilateral shift $S_N$ of finite multiplicity $N$ are studied. It is proved that…

We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to $0$ on the positive cone. Some results on unbounded…