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P. Sam Johnson

Publications and source records attributed to P. Sam Johnson.

At least 19 recordsLinked to original sources
arXiv2026

Characterization of Frame-Related Sequences on Semi-Hilbert Spaces

Let $A\in\mathcal{B}(H)^+$ and $B\in\mathcal{B}(\ell^2)^+$ be positive bounded operators. We investigate reduced weighted adjoints of densely defined closable operators and use them to develop frame-type systems in the semi-Hilbert spaces induced by $A$ and $B$. Closedness, boundedness, range behavior, and algebraic properties of the $A$-adjoint are established, with particular attention to sums, products, and double adjoints in the unbounded setting. We then study the associated $AB$-analysis, $AB$-synthesis, and $AB$-frame operators. Operator-theoretic characterizations of $AB$-Bessel sequences, $AB$-lower semi-frames, and $AB$-frames are obtained, and examples show that several natural weighted-adjoint identities may hold only as proper inclusions. A Douglas-type range characterization for $AB$-frames is also derived.

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arXiv2026

Norm Inequalities for Complementable Operators and Parallel Sums

This paper investigates the structural and quantitative behaviors of complementable operators on Hilbert spaces, focusing on their norm characteristics and geometric profiles. We establish a comprehensive framework of norm inequalities and lower-bound relationships between a bounded linear operator and its generalized Schur complement (bilateral shorted operator). Under explicit operator factorization and range inclusion criteria, we define the exact conditions under which a bounded linear operator contracts or expands vectors relative to its Schur complement. Furthermore, we explore the lower boundedness and stability configurations of $(M, N, \lambda)$-complementable operators, proving that a bounded-below Schur complement acts as a sufficient condition to propagate injectivity and lower-bounded stability to the global operator. These structural results are subsequently applied to the network-theoretic setting of the parallel sum of two bounded linear operators. With some specific orthogonality conditions, we derive a novel norm decomposition identity, sharp two-sided global operator bounds, and algebraic restrictions on Douglas reduced solutions via Moore-Penrose inverses.

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arXiv2025

On unbounded complementable operators

The concept of complementability is extended from bounded operators to densely defined operators on Hilbert spaces. By introducing appropriate projections and decomposition techniques, a framework is developed for analyzing complementability in this broader context. The results provide new insights into the structure of unbounded operators, contributing to the ongoing development of operator theory.

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arXiv2025

Direct Sum of Lower Semi-Frames in Hilbert Spaces

In this paper, structural properties of lower semi-frames in separable Hilbert spaces are explored with a focus on transformations under linear operators (may be unbounded). Also, the direct sum of lower semi-frames, providing necessary and sufficient conditions for the preservation of lower semi-frame structure, is examined.

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arXiv2024

Riesz Bases in Krein Spaces

We start by introducing and studying the definition of a Riesz basis in a Krein space $(\mathcal{K},[.,.])$, along with a condition under which a Riesz basis becomes a Bessel sequence. The concept of biorthogonal sequence in Krein spaces is also introduced, providing an equivalent characterization of a Riesz basis. Additionally, we explore the concept of the Gram matrix, defined as the sum of a positive and a negative Gram matrices, and specify conditions under which the Gram matrix becomes bounded in Krein spaces. Further, we characterize the conditions under which the Gram matrices $\{[f_n,f_j]_{n,j \in I_+}\}$ and $\{[f_n,f_j]_{n,j \in I_-}\}$ become bounded invertible operators. Finally, we provide an equivalent characterization of a Riesz basis in terms of Gram matrices.

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arXiv2024

On the generalized Cauchy dual of closed operators in Hilbert spaces

In this paper, we introduce the generalized Cauchy dual $w(T) = T(T^{*}T)^{\dagger}$ of a closed operator $T$ with the closed range between Hilbert spaces and present intriguing findings that characterize the Cauchy dual of $T$. Additionally, we establish the result $w(T^{n}) = (w(T))^{n}$, for all $n \in \mathbb{N}$, where $T$ is a quasinormal EP operator.

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arXiv2024

Convergence of Complementable Operators

Complementable operators extend classical matrix decompositions, such as the Schur complement, to the setting of infinite-dimensional Hilbert spaces, thereby broadening their applicability in various mathematical and physical contexts. This paper focuses on the convergence properties of complementable operators, investigating when the limit of sequence of complementable operators remains complementable. We also explore the convergence of sequences and series of powers of complementable operators, providing new insights into their convergence behavior. Additionally, we examine the conditions under which the set of complementable operators is the subset of set of boundary points of the set of non-complementable operators with respect to the strong operator topology. The paper further explores the topological structure of the subset of complementable operators, offering a characterization of its closed subsets.

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arXiv2024

Characterizations of closed EP operators on Hilbert spaces

In this paper, we present intriguing findings that characterize both the closed (unbounded) and bounded EP operators on Hilbert spaces. Additionally, we demonstrate the result $\gamma(T) \leq r(T)$, where $T$ is a bounded EP operator, and $\gamma(T) \text{ and } r(T)$ represent the reduced minimum modulus and the spectral radius of $T$, respectively.

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arXiv2024

Frame Scaling by Graphs

In this paper, we investigate the scalability of a given frame in $\mathbb{R}^n$ by using graphs. For each frame $\phi$ in $\mathbb{R}^n$, we associate a simple undirected graph $G(\phi)$ and use it to verify the scalability of $\phi$. We provide some necessary conditions to test the scalability of a given frame. Finally, we study the scalability of some special classes of frames by using graphs.

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arXiv2024

Complementable Operators and their Schur Complements

In this paper, we characterize complementable operators and provide more precise expressions for the Schur complement of these operators using a single Douglas solution. We demonstrate the existence of subspaces where the given operator is invariably complementable. Additionally, we investigate the range-Hermitian property of these operators.

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arXiv2024

Hyers-Ulam Stability of Unbounded Closable Operators in Hilbert Spaces

In this paper, we discuss the Hyers-Ulam stability of closable (unbounded) operators with several interesting examples. We also present results pertaining to the Hyers-Ulam stability of the sum and product of closable operators to have the Hyers-Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of $2 \times 2$ block matrix $\mathcal A$ in order to have the Hyers-Ulam stability.

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arXiv2024

$A$-approximate point spectrum of $A$-bounded operators in semi-Hilbertian spaces

This paper delves into several characterizations of $A$-approximate point spectrum of A-bounded operators acting on a complex semi-Hilbertian space $H$ and also investigates properties of the $A$-approximate point spectrum for the tensor product of two $A^{\frac{1}{2}}$-adjoint operators. Furthermore, several properties of $A$-normal operators have been established.

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arXiv2024

A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces

In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |\lambda|: \lambda \in \sigma_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a closed range) and $r_{A}(T)$ is the $A$-spectral radius of $T$ in $B_{A^{\frac{1}{2}}}(H)$. Also we prove that $\sup \{ |\lambda|: \lambda \in \sigma_{A}(T)\} = r_{A}(T), \text{ when } T \in B_{A^{\frac{1}{2}}}(H) \text { commutes with } A$. By introducing $A$-Harte spectrum $\sigma_{A_{h}}(\mathbf{T})$ of a $d$-tuple operator $\mathbf{T}= (T_{1},\dots,T_{d}) \in (B_{A^{\frac{1}{2}}}(H))^{d}$, we prove that $r_{A_{h}}(\mathbf{T}) \leq \sup \{\|\lambda\|_{2}: \lambda \in \sigma_{A_{h}}(\mathbf{T})\}$, where $r_{A_{h}}(\mathbf{T})$ is the $A$-Harte spectral radius of $\mathbf{T}$.

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arXiv2022

Lipschitz p-Approximate Schauder Frames

With the aim of representing subsets of Banach spaces as an infinite series using Lipschitz functions, we study a variant of metric frames which we call Lipschitz p-approximate Schauder frames (Lipschitz p-ASFs). We characterize Lipschitz p-ASFs and their duals completely using the canonical Schauder basis for classical sequence spaces. Similarity of Lipschitz p-ASF is introduced and characterized.

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arXiv2022

Controlled continuous $g$-frames and their duals in Hilbert spaces

In this paper, we characterize and study the concept of controlled continuous $g$-frame which is an extension of continuous $g$-frame in Hilbert spaces. We introduce the concept of controlled continuous dual $g$-frame and observe some interesting properties of it. Finally, we characterize all controlled continuous dual $g$-frames of a controlled continuous $g$-frame.

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arXiv2022

Reverse Order Law for Closed Range Operators in Hilbert Spaces

We present more than 50 results including some range inclusion results to characterize reverse order law for Moore-Penrose inverse of closed range Hilbert space operators. We use basic properties of Moore-Penrose inverse to prove the results. Some examples are also provided to illustrate failure cases to hold the reverse order law in infinite dimensional settings.

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arXiv2022

P-Operators on Hilbert Spaces

A real square matrix $A$ is called a P-matrix if all its principal minors are positive. Using the sign non-reversal property of matrices, the notion of P-matrix has been recently extended by Kannan and Sivakumar to infinite-dimensional Banach spaces relative to a given Schauder basis. Motivated by their work, we discuss P-operators on separable real Hilbert spaces. We also investigate P-operators relative to various orthonormal bases.

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