Hilbert Space Operators Research Articles

A note on the maximal numerical range of operators for a sesquilinear form

In this paper, we investigate the concept of maximal numerical range of operators relative to a sesquilinear form which generalizes the classical maximal numerical range of operators in Hilbert spaces and we show some structural properties of these sets.

Operator inequalities in reproducing kernel Hilbert spaces

In this paper, by using some classical Mulholland type inequality, Berezin symbols and reproducing kernel technique, we prove the power inequalities for the Berezin number $ber(A)$ for some self-adjoint operators $A$ on ${H}(\Omega )$. Namely, some Mulholland type inequality for reproducing kernel Hilbert space operators are established. By applying this inequality, we prove that $(ber(A))^{n}\leq C_{1}ber(A^{n})$ for any positive operator $A$ on ${H}(\Omega )$.

Strong correlation, Bloch bundle topology, and spinless Haldane–Hubbard model

Different realizations of the Hubbard operators in different Hilbert spaces give rise to various microscopic lattice electron models driven by strong correlations. In terms of the Gutzwiller projected electron operators, the most familiar examples are the conventional t−J model and the BCS-Hubbard model at strong coupling. Here we focus on the spin-dopon representation of the Hubbard operators. In this case the no double occupancy (NDO) constraint can be reexpressed as a Kondo interaction. As an explicit example, the effective low-energy action is derived in terms of itinerant spinless fermions (dopons) strongly interacting with localized lattice spins. The spontaneous breaking of the time-reversal symmetry reduces that action to one describing a spinless version of the Haldane–Hubbard topological theory. This result suggests that the topologically nontrivial U(1) Bloch bundle associated with this model can be realized dynamically due to the presence of strong correlations even in the absence of any external local or global fluxes.

Random fixed-point algorithms for the stochastic split common problem of firmly non-expansive operators

Three fixed-point equations for solving both split common and feasibility problems of firmly non-expansive operator in a stochastic domain was studied. Stochastic convergence of the iterations based on random-type iterative scheme for solving stochastic split common and feasibility problems was also established. The mapping of the operators in Hilbert space was adopted.

Some A-spectral radius inequalities for A-bounded Hilbert space operators

Let $$r_A(T)$$ denote the A-spectral radius of an operator T which is bounded with respect to the seminorm induced by a positive operator A on a complex Hilbert space $${\mathcal {H}}$$ . In this paper, we aim to establish several A-spectral radius inequalities for products, sums and commutators of A-bounded operators. Some applications of our results are provided. Moreover, we give an affirmative answer to the question recently posed by Baklouti and Namouri (Banach J Math Anal 16:12, https://doi.org/10.1007/s43037-021-00167-1 , 2022) regarding the connection between the notions of A-spectral radius and A-spectrum for A-bounded operators.

On the nonlinear perturbations of self-adjoint operators

Abstract Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=N\left(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {\mathcal{ {\mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

English

In this article, we prove an inner product inequality for Hilbert space operators. This inequality, then, is utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain the refined versions.

On solving variational inequality problems involving quasimonotone operators via modified Tseng's extragradient methods with convergence analysis

The main objective of this research is to find the numerical solution of variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces. The main advantage of these iterative schemes is that they allow the uncomplicated calculation of step size rules that depend on the knowledge of an operator explanation instead of the Lipschitz constant or some other line search method. The proposed iterative schemes follow a monotone and non-monotone step size procedure based on mapping (operator) information as a replacement for its Lipschitz constant or some other line search method. The strong convergences are well proven, analogous to the proposed methods, and impose certain control specification conditions. Finally, to verify the effectiveness of the iterative methods, we present some numerical experiments.

Core–EP orthogonal operators

The concept of the core orthogonality (CO) was recently introduced for two square matrices A and B of the same size and of index at most 1 as follows: A is core orthogonal to B if and , where is the core inverse of A. Using the core–EP inverse instead of the core inverse, we extend the concept of the CO and present the new concept of the core–EP orthogonality (CEPO) for two Drazin invertible bounded linear Hilbert space operators A and B as follows: A is said to be core–EP orthogonal to B if and , where is the core–EP inverse of A. A number of characterizations for CEPO are presented as well as the relation between the CEPO and core–EP additivity. Applying CEPO, the concept of strong core–EP orthogonality is defined and characterized.

A hyperbolic universal operator commuting with a compact operator

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and which commutes with a quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a “hyperbolic” composition operator.

Some Gap Relations Between Operator Norm with Spectral and Numerical Radii of Direct Sum Hilbert Space Operators

Some Gap Relations Between Operator Norm with Spectral and Numerical Radii of Direct Sum Hilbert Space Operators

On left and right west-stampfli decomposition

In this paper we define and investigate the decomposition of a Hilbert space operator T in the form T = K+Q where K is a compact and the approximate points spectrum (or the surjectivity spectrum) of Q is identical to the set of all accumulation point of the approximate point spectrum ( or the surjectivity spectrum) of T. Also, we provide the relation between operators having these decomposition and left (or right) Stampfli operators.

On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

Ontological Revision and Quantum Mechanics

We observe that there exists a presumed ontology, on the basis of which a dynamical theory is formulated. Extracting the essence of the ontology underlying classical mechanics, we revise it suitably to accommodate quantum mechanics seamlessly. The revision goes through two stages: First, we elaborate the concept of “state” as the propensity of an object to arouse events on discerners. Second, we integrate the position and momentum spaces into a single entity, related via the Fourier transform. Based on this revised ontology, quantum mechanics is deduced naturally without relying upon any artificial assumptions like the “quantum condition” or mathematical apparatuses like the Hilbert space and self-adjoint operators. In consequence, such thorny issues as the “measurement problem” are shown to either resolve or simply disappear.

Self-adaptive algorithms for the split problem of the quasi-pseudocontractive operators in Hilbert spaces

<abstract><p>Self-adaptive algorithms are presented for solving the split common fixed point problem of quasi-pseudocontractive operators in Hilbert spaces. Weak and strong convergence theorems are given under some mild assumptions.</p></abstract>

Управляемые g-атомарные подпространства для операторов в гильбертовых пространствах

Управляемые g-атомарные подпространства для операторов в гильбертовых пространствах

The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behavior of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composite operator decay faster than those of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

Iterative solutions via some variants of extragradient approximants in Hilbert spaces

<abstract><p>This paper provides iterative solutions, via some variants of the extragradient approximants, associated with the pseudomonotone equilibrium problem (EP) and the fixed point problem (FPP) for a finite family of $ \eta $-demimetric operators in Hilbert spaces. The classical extragradient algorithm is embedded with the inertial extrapolation technique, the parallel hybrid projection technique and the Halpern iterative methods for the variants. The analysis of the approximants is performed under suitable set of constraints and supported with an appropriate numerical experiment for the viability of the approximants.</p></abstract>

Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

The stability of the solution to the equation $(*)\dot{u} = F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We assume that the Fr\'echet derivative of $F(t,u)$ is H\"{o}lder continuous of order $q>0$ with respect to $u$ for any fixed $t\ge 0$, i.e., $\|F'_u(t,w) - F'_u(t,v)\|\le \alpha(t)\|v - w\|^{q}$, $q>0$. We proved that the equilibrium solution $v=0$ to the equation $\dot{v} = F(t,v)$ is Lyapunov stable under persistently acting perturbation $f(t)$ if $\sup_{t\ge 0}\int_0^t \alpha(\xi)\|U(t,\xi)\|\, d\xi<\infty$ and $\sup_{t\ge 0}\|U(t)\|<\infty$. Here, $U(t):=U(t,0)$ and $U(t,\xi)$ is the solution to the equation $\frac{d}{dt}{U}(t,\xi) = F'_u(t,0)U(t,\xi)$, $t\ge \xi$, $U(\xi,\xi)=I$, where $I$ is the identity operator in $\mathcal{X}$. Sufficient conditions for the solution $u(t)$ to equation (*) to be bounded and for $\lim_{t\to\infty}u(t) = 0$ are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.

CONTROLLABILITY OF FREDHOLM’S INTEGRO-DIFFERENTIAL EQUATIONS WITH BY A DEGENERATE KERNEL IN HILBERT SPACES

The work examines integro-differential equations Fredholm with a degenerate kernel with Hilbert control spaces. The need to study these equations is related to numerous ones applications of integro- differential equations in mathematics, physics, technology, economy and other fields. Complexity the study of integro-differential equations is connected with the fact that the integral-differential operator is not solvable everywhere. There are different approaches to the solution of not everywhere solvable linear operator equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc. The problem of obtaining coefficient conditions of solvability and analytical presentation of general solutions of integro-differential equations is a rather difficult problem, so frequent solutions will suffice are obtained by numerical methods. In this connection, Fredholm’s integro-differential equations with degenerate kernel and control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-differential equations with a degenerate kernel in abstract Hilbert spaces. As an intermediate result in the work using the results of pseudoinversion of integral operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-differential equations without control in the abstract Hilbert spaces. To establish the controllability criterion is not solvable everywhere integro-differential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used significantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces. With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-differential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.