Operators In Spaces Research Articles

Fuzzy Orthogonality of Operators: A Framework in Fuzzy Banach Spaces

In this paper, we introduce a new notion of fuzzy orthogonality between two linear operators on a fuzzy Banach space. We demonstrate that if the fuzzy norm satisfies the condition (N6), then the operators are orthogonal in the Birkhoff–James sense with respect to the [Formula: see text]-norm. Additionally, we establish several fundamental results concerning fuzzy orthogonality of linear operators in finite-dimensional fuzzy Banach spaces.

$$\textit{A}$$-approximate point spectrum of $$\textit{A}$$-bounded operators in semi-Hilbertian spaces

$$\textit{A}$$-approximate point spectrum of $$\textit{A}$$-bounded operators in semi-Hilbertian spaces

Topological structure of the space of composition operators on the Hardy space of Dirichlet series

Topological structure of the space of composition operators on the Hardy space of Dirichlet series

A functional representation approach to vector lattice covers for spaces of compact operators

A functional representation approach to vector lattice covers for spaces of compact operators

Bosonic fortuity in vector models

We investigate the space of U(N) gauge-invariant operators in coupled matrix-vector systems at finite N, extending previous work on single matrix models. By using the Molien-Weyl formula, we compute the partition function and identify the structure of primary and secondary invariants. In specific examples we verify, using the trace relations, that these invariants do indeed generate the complete space of gauge invariant operators. For vector models with f ≤ N species of vectors, the space is freely generated by primary invariants, while for f > N, secondary invariants appear, reflecting the presence of nontrivial trace relations. We derive analytic expressions for the number of secondary invariants and explore their growth. These results suggest a bosonic analogue of the fortuity mechanism. Our findings have implications for higher-spin holography and gauge-gravity duality, with applications to both vector and matrix models.

Sequential learning on a tensor network Born machine with trainable token embedding

Abstract Generative models aim to learn the probability distributions underlying data, enabling the generation of new, realistic samples. Quantum-inspired generative models, such as Born machines based on the matrix product state (MPS) framework, have demonstrated remarkable capabilities in unsupervised learning tasks. This study advances the Born machine paradigm by introducing trainable token embeddings through positive operator-valued measurements (POVMs), replacing the traditional approach of static tensor indices. Key technical innovations include encoding tokens as quantum measurement operators with trainable parameters and leveraging QR decomposition to adjust the physical dimensions of the MPS. This approach maximizes the utilization of operator space and enhances the model’s expressiveness. Empirical results on RNA data demonstrate that the proposed method significantly reduces negative log-likelihood (NLL) compared to one-hot embeddings, with higher physical dimensions further enhancing single-site probabilities and multi-site correlations. The model also outperforms GPT-2 in single-site estimation and achieves competitive correlation modeling, showcasing the potential of trainable POVM embeddings for complex data correlations in quantum-inspired sequence modeling.

Mean Ergodicity of Multiplication Operators in Weighted Dirichlet Spaces

We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Cesàro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Cesàro boundedness for multiplication operators. We perform the same study for adjoints of multiplication operators. As a particular example, we obtain a uniform mean ergodic multiplication operator in Dirichlet spaces that fails to be power bounded.

The m-weak group orthogonality for operators

The main goal is extending the concept of the core–EP orthogonality to the m-weak group orthogonality for bounded linear Drazin invertible Hilbert space operators, using the m-weak group inverse. Different properties and characterizations of m-weak group orthogonal operators are proved as well as their operator matrix forms. The connection between the m-weak group binary relation and the m-weak group orthogonality is given. We also study additive properties for the m-weak group inverse. Consequently, we study the weak group orthogonality for operators.

BV formalism and partition functions

The BV formalism is a well-established method for analyzing symmetries and quantizing field theories. In this paper, we use BV formalism to derive partition functions and the space of gauge invariant operators implementing the equations of motions and their redundancies for selected theories. We discuss various interpretations of the results, some dualities, and relation to the first quantized models.

On the Symbols of Strictly m-Null Elementary Operators

This paper extends the previous work by the author on m-null pairs of operators in Hilbert space. If an elementary operator L has elementary symbols A and B that are p-null and q-null, respectively, then L is (p+q−1)-null. Here, we prove the converse under strictness conditions, modulo some nonzero multiplicative constant—if L is strictly (p+q−1)-null, then a scalar λ≠0 exists such that λA is strictly p-null and λ−1B is strictly q-null. Our constructive argument relies essentially on algebraic and combinatorial methods. Thus, the result obtained by Gu on m-isometries is recovered without resorting to spectral analysis. For several operator classes that generalize m-isometries and are subsumed by m-null operators, the result is new.

F-Alternate Interpolative Ciric-Reich-Rus Contraction Mapping Theorem

In [1], the authors introduced the interpolative Ciric-Reich-Rus operator in Branciari metric space and obtained some fixed point theorems. In [2], an alternate characterization of the interpolative Ciric-Reich-Rus operator was given, and some fixed point theorems were obtained. In the present paper, we consider the alternate interpolative Ciric-Reich-Rus operator is an F-contraction [3], and obtain a fixed point theorem.

Spectral Analysis of Nonlinear Operators: Theory and Applications to Neural Networks and Optimization

This paper presents a nonlinear spectral framework for analyzing monotone and nonexpansive operators in Banach and Hilbert spaces. We construct a nonlinear spectral resolution for maximal monotone operators using Yosida approximations and Fitzpatrick functions, leading to a family of nonlinear projections and an associated spectral measure. For nonexpansive mappings, we establish an iterative spectral approximation based on Krasnoselskii iterations, with proven convergence and recovery of nonlinear eigenvectors. We further extend this framework to ReLU-based neural networks, analyzing spectral bounds, depth-dependent scaling, and gradient alignment. These results bridge nonlinear operator theory and neural architectures, offering new tools for theoretical analysis and applications in optimization, physics, and machine learning.

On the Cauchy problem for the Langevin-type fractional equation

In this article, the Cauchy problem for the Langevin-type time-fractional equation Dβ t (Dtαu(t)) + Dtβ(Au(t)) = f(t); (0 < t ≤ T) is studied. Here α; β 2 (0; 1), Dtα; Dtβ is the Caputo derivative and A is an unbounded self-adjoint operator in a separable Hilbert space. Under certain conditions, we establish the existence and uniqueness of the solution and provide an explicit representation of it using eigenfunction expansions

Norm-Attainable Operators in Hilbert Spaces: Probabilistic and Finite-Rank Perspectives

On this note, we investigate norm-attainable operators in Hilbert spaces, focusing on probabilistic and finite-rank perspectives. We present key results concerning the existence and properties of norm-attaining vectors, particularly for compact and finite-rank operators. Using spectral theory and concentration of measure, we show that norm-attaining vectors form compact subspaces in the unit sphere. Additionally, we explore how unitary transformations affect these vectors and discuss the implications for operator theory and functional analysis.

Nonlinear composition operators in bvp-spaces: acting conditions and boundedness

The aim of this paper is to give the answer to the problem of characterization of acting conditions (necessary as well as sufficient) for nonlinear composition operators in some sequence spaces. We also characterize their boundedness and local boundedness. We focus on nonlinear composition operators acting to or from the space bvp(E) of all sequences of p-bounded variation; here p≥1 and E is a normed space.

Some counterexamples in the theory of linear operators

Counterexamples are given that show significant differences between classes of linear operators in topological vector spaces.

Generalized A-numerical radius inequalities in semi-Hilbert spaces through the angle between two vectors

This paper presents several inequalities related to the A-numerical radius in the context of semi-Hilbert space operators. By integrating modern techniques from operator theory and functional analysis, we derive new inequalities for the A-numerical radius that emphasize the unique characteristics of semi-Hilbert space operators. Among other results, it is shown that, if , where A is a positive and onto operator, and T has a polar decomposition given by T = U|T|, and , then where either for all unit vectors for all A-unit vectors .

Basis property of the root functions of a class of nonlocal differential operators

In this work, we utilize the perturbation theory of self-adjoint unbounded operators in Hilbert spaces to establish the basis properties of a broad class of fractional ordinary differential equations (ODEs) involving both left- and right-sided nonlocal derivatives, including Riemann-Liouville or Caputo derivatives. We demonstrate that the system of root functions associated with the spectral problem forms a Riesz basis with brackets. Finally, we discuss practical numerical schemes for approximating eigenvalues and eigenfunctions based on these theoretical findings.

Substochastic operators in symmetric spaces

Abstract First, we solve a crucial problem under which conditions increasing uniform ‐monotonicity is equivalent to lower local uniform ‐monotonicity. Next, we investigate properties of substochastic operators on with applications. Namely, we show that a countable infinite combination of substochastic operators is also substochastic. Using ‐monotonicity properties, we prove several theorems devoted to the convergence of the sequence of substochastic operators in the norm of a symmetric space under addition assumption on . In our final discussion, we focus on compactness of admissible operators for Banach couples under additional assumption.

Novel p-numerical radius inequalities for Hilbert space operators

Novel p-numerical radius inequalities for Hilbert space operators