arXiv subjects

Soumyashant Nayak

Publications and source records attributed to Soumyashant Nayak.

14 recordsLinked to original sources

arXiv2026

On the convergence of the normalized power sequence of Riesz operators

Let $\mathscr{H}$ be a complex Hilbert space and $\mathscr{B}(\mathscr{H})$ be the algebra of all bounded linear operators on $\mathscr{H}$. For $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. We refer to the sequence $\big\{|T^n|^{\frac{1}{n}}\big\}_{n \in \mathbb{N}}$ as the NPS (normalized power sequence) of $T$. In this article, we show that the NPS of a Riesz operator $R \in \mathscr{B}(\mathscr{H})$ converges in norm to a positive operator $H$, and provide an explicit description of the spectral resolution of $H$ in terms of the Riesz idempotents associated with the non-zero eigenvalues of $R$. Since every compact operator is a Riesz operator, this gives us a stronger, spatial generalization of the Yamamoto-Davis theorem, which asserts that $\lim_{n \to \infty} s_j(K^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of the compact operator $K$, where $s_j(\cdot)$ denotes the $j^{\textrm{th}}$-largest singular value. In recent work, the present authors have established the norm convergence of the NPS for spectral operators. Using a rank-one perturbation of a unitary operator, we demonstrate that this fails in general for essentially spectral operators (of which Riesz operators form a subclass).

math.FA↗

arXiv2025

On the convergence of the normalized power sequence of spectral operators on Hilbert space

Let $\mathscr{H}$ be a complex Hilbert space, and let $\mathscr{B}(\mathscr{H})$ denote the set of all bounded operators on $\mathscr{H}$ . For an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{\frac{1}{n}} \}_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $II_1$ factor.

math.FA↗

arXiv2025

On the Jordan-Chevalley-Dunford decomposition of operators in type $I$ Murray-von Neumann algebras

We show that, for $n \ge 3$, the mapping on $M_n(\mathbb{C})$ which sends a matrix to its diagonalizable part in its Jordan-Chevalley decomposition, is {\bf norm-unbounded} on any neighbourhood of the zero matrix. Let $X$ be a Stonean space, and $\mathcal{N}(X)$ denote the $*$-algebra of (unbounded) normal functions on $X$, containing $C(X)$ as a $*$-subalgebra. We show that every element of $M_n\big(\mathcal{N}(X)\big)$ has a unique Jordan-Chevalley decomposition. Furthermore, when $n \ge 3$ and $X$ has infinitely many points, using the unboundedness of the Jordan-Chevalley decomposition, we show that there is an element of $M_n\big(C(X)\big)$ whose diagonalizable and nilpotent parts are not bounded, that is, do not lie in $M_n\big(C(X)\big)$. Using these results in the context of a type $I$ finite von Neumann algebra $\mathscr{N}$, we prove a canonical Jordan-Chevalley-Dunford decomposition for densely-defined closed operators affiliated with $\mathscr{N}$, expressing each such operator as the strong-sum of a unique commuting pair consisting of (what we call) a $\mathfrak{u}$-scalar-type affiliated operator and an $\mathfrak{m}$-quasinilpotent affiliated operator. The functorial nature of Murray-von Neumann algebras, coupled with the above observations, indicates that considering unbounded affiliated operators is both necessary and natural in the quest for a Jordan-Chevalley-Dunford decomposition for bounded operators in type $II_1$ von Neumann algebras.

math.OA↗

arXiv2024

Algebraic aspects and functoriality of the set of affiliated operators

In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let $\mathscr{M}_{\text{aff}}$ denote the set of unbounded operators of the form $T = AB^{\dagger}$ for $A, B \in \mathscr{M}$ with $\ker(B)\subseteq\ker(A)$ , where $(\cdot)^{\dagger}$ denotes the Kaufman inverse. We show that $\mathscr{M}_{\text{aff}}$ is closed under product, sum, Kaufman-inverse and adjoint, and has the structure of a right near-semiring; Moreover, the above quotient representation of an operator in $\mathscr{M}_{\text{aff}}$ is essentially unique. The Murray-von Neumann affiliated operators for $\mathscr{M}$ turn out to be precisely the closed operators in $\mathscr{M}_{\text{aff}}$. Let $Φ$ be a unital normal homomorphism between represented von Neumann algebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. With the help of the quotient representation, we obtain a canonical extension of $Φ$ to a mapping $Φ_{\text{aff}} : \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}$ which respects sum, product, Kaufman-inverse, and adjoint. Thus $\mathscr{M}_{\text{aff}}$ is intrinsically associated with $\mathscr{M}$ and transforms functorially as we change representations of $\mathscr{M}$. Furthermore, $Φ_{\text{aff}}$ preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein-von Neumann extensions of densely-defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via `abstract nonsense'.

math.OA↗

arXiv2023

A stronger form of Yamamoto's theorem on singular values

For a matrix $T \in M_m(\mathbb{C})$, let $|T| : = \sqrt{T^*T}$. For $A \in M_m(\mathbb{C})$, we show that the matrix sequence $\big\{ |A^n|^{\frac{1}{n}} \big\}_{n \in \mathbb{N}}$ converges in norm to a positive-semidefinite matrix $H$ whose $j^{\textrm{th}}$-largest eigenvalue is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of $A$ (for $1 \le j \le m$). In fact, we give an explicit description of the spectral projections of $H$ in terms of the eigenspaces of the diagonalizable part of $A$ in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that $\lim_{n \to \infty} s_j(A^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of $A$, where $s_j(A^n)$ denotes the $j^{\textrm{th}}$-largest singular value of $A^n$. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, $t \mapsto e^{tA}$.

math.FA↗

arXiv2022

On Murray-von Neumann algebras -- I: Topological, Order-Theoretic and Analytical Aspects

For a countably decomposable finite von Neumann algebra $\mathscr{R}$, we show that any choice of a faithful normal tracial state on $\mathscr{R}$ engenders the same measure topology on $\mathscr{R}$ in the sense of Nelson (J. Func. Anal., 15 (1974), 103--116). Consequently it is justified to speak of `the' measure topology of $\mathscr{R}$. Having made this observation, we extend the notion of measure topology to general finite von Neumann algebras and denominate it the $\mathfrak{m}$-topology. We note that the procedure of $\mathfrak{m}$-completion yields Murray-von Neumann algebras in a functorial manner and provides them with an intrinsic description as unital ordered complex topological $*$-algebras. This enables the study of abstract Murray-von Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes apparent the appropriate notion of Murray-von Neumann subalgebras, and the intrinsic nature of the spectrum and point spectrum of elements, independent of their ambient Murray-von Neumann algebra. In this context, we show the well-definedness of the Borel function calculus for normal elements and use it along with approximation techniques in the $\mathfrak{m}$-topology to transfer many standard operator inequalities involving bounded self-adjoint operators to the setting of (unbounded) self-adjoint operators in Murray-von Neumann algebras. On the algebraic side, Murray-von Neumann algebras have been described as the Ore localization of finite von Neumann algebras with respect to their corresponding multiplicative subset of non-zero-divisors. Our discussion reveals that, in addition, there are fundamental topological, order-theoretic and analytical facets to their description.

math.OA↗

arXiv2022

On products of symmetries in von Neumann algebras

Let $\mathscr{R}$ be a type $II_1$ von Neumann algebra. We show that every unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in $\mathscr{R}$ with finite spectrum may be decomposed as the product of four symmetries in $\mathscr{R}$. Consequently, the set of products of four symmetries in $\mathscr{R}$ is norm-dense in the unitary group of $\mathscr{R}$. Furthermore, we show that the set of products of three symmetries in a von Neumann algebra $\mathscr{M}$ is not norm-dense in the unitary group of $\mathscr{M}$. This strengthens a result of Halmos-Kakutani which asserts that the set of products of three symmetries in $\mathcal{B}(\mathscr{H})$, the ring of bounded operators on a Hilbert space $\mathscr{H}$, is not the full unitary group of $\mathcal{B}(\mathscr{H})$.

math.OA↗

arXiv2021

A framework for rank identities -- With a view towards operator algebras

For every square matrix $A$ over a field $\mathbb{K}$, we have the equality $\mathrm{rank}(A) + \mathrm{rank}(I-A) = \mathrm{rank}(I) + \mathrm{rank}(A-A^2)$ where $I$ denotes the identity matrix with the same dimensions as $A$. In this article, we start a program to systematically characterize and generalize such rank identities with a view towards applications to operator algebras. We initiate the study of so-called ranked rings (unital rings with a `rank-system'), the main examples of interest being finite von Neumann algebras, Murray-von Neumann algebras, and von Neumann rank-rings. In our framework, a field $\mathbb{K}$ may be viewed as a ranked ring with a $\mathbb{Z}^+$-valued rank-system consisting of the usual rank functions on $M_n(\mathbb{K})$ (for $n \in \mathbb{N})$ and serves as the motivating example. We show that a finite von Neumann algebra $\mathscr{R}$ with center $\mathscr{C}$ (and the corresponding Murray-von Neumann algebra $\mathscr{R}_{\textrm{aff}}$) may be endowed with a $\mathscr{C}^{+}$-valued rank-system, considering $\mathscr{C}^{+}$ as a commutative monoid with respect to operator addition. We give an algorithm to generate rank identities in ranked $\mathbb{K}$-algebras via the polynomial function calculus, and in ranked complex Banach algebras via the holomorphic function calculus. As an illustrative application, using these abstract rank identities we show that the sum of finitely many idempotents $e_1, \ldots, e_m$ in a finite von Neumann algebra is an idempotent if and only if they are mutually orthogonal, that is, $e_i e_j = δ_{ij} e_i$ for $1 \le i, j \le m$. In contrast, this does not hold in general in the ring of bounded operators on an infinite-dimensional complex Hilbert space.

math.RA↗

arXiv2021

The Douglas lemma for von Neumann algebras and some applications

In this article, we discuss some applications of the well-known Douglas factorization lemma in the context of von Neumann algebras. Let $\mathcal{B}(\mathscr{H})$ denote the set of bounded operators on a complex Hilbert space $\mathscr{H}$, and $\mathscr{R}$ be a von Neumann algebra acting on $\mathscr{H}$. We prove some new results about left (or, one-sided) ideals of von Neumann algebras; for instance, we show that every left ideal of $\mathscr{R}$ can be realized as the intersection of a left ideal of $\mathcal{B}(\mathscr{H})$ with $\mathscr{R}$. We also generalize a result by Loebl and Paulsen (Linear Algebra Appl. 35 (1981), 63--78) pertaining to $C^*$-convex subsets of $\mathcal{B}(\mathscr{H})$ to the context of $\mathscr{R}$-bimodules.

math.OA↗

arXiv2020

Matrix algebras over algebras of unbounded operators

Let $\mathscr{M}$ be a $II_1$ factor acting on the Hilbert space $\mathscr{H}$, and $\mathscr{M}_{\textrm{aff}}$ be the Murray-von Neumann algebra of closed densely-defined operators affiliated with $\mathscr{M}$. Let $τ$ denote the unique faithful normal tracial state on $\mathscr{M}$. By virtue of Nelson's theory of non-commutative integration, $\mathscr{M}_{\textrm{aff}}$ may be identified with the completion of $\mathscr{M}$ in the measure topology. In this article, we show that $M_n(\mathscr{M}_{\textrm{aff}}) \cong M_n(\mathscr{M})_{\textrm{aff}}$ as unital ordered complex topological $*$-algebras with the isomorphism extending the identity mapping of $M_n(\mathscr{M}) \to M_n(\mathscr{M})$. Consequently, the algebraic machinery of rank identities and determinant identities are applicable in this setting. As a step further in the Heisenberg-von Neumann puzzle discussed by Kadison-Liu (SIGMA, 10 (2014), Paper 009), it follows that if there exist operators $P, Q$ in $\mathscr{M}_{\textrm{aff}}$ satisfying the commutation relation $Q \; \hat \cdot \; P \; \hat - \; P \; \hat \cdot \; Q = {i\mkern1mu} I$, then at least one of them does not belong to $L^p(\mathscr{M}, τ)$ for any $0 < p \le \infty$. Furthermore, the respective point spectrums of $P$ and $Q$ must be empty. Hence the puzzle may be recasted in the following equivalent manner - Are there invertible operators $P, A$ in $\mathscr{M}_{\textrm{aff}}$ such that $P^{-1} \; \hat \cdot \; A \; \hat \cdot \; P = I \; \hat + \; A$? This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in $\mathscr{M}_{\textrm{aff}}$ in an essential way.

math-ph↗

arXiv2018

Jensen's inequality in finite subdiagonal algebras

Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal tracial state $τ$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathscr{M}$ with respect to a $τ$-preserving faithful normal conditional expectation $Φ$ on $\mathscr{M}$. Let $Δ$ denote the Fuglede-Kadison determinant corresponding to $τ$. For $X \in \mathscr{M}$, define $|X| := (X^*X)^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen's inequality for finite subdiagonal algebras i.e. $Δ(Φ(A)) \le Δ(A)$ for an operator $A \in \mathfrak{A}$, thus resolving a long-standing open problem posed by Arveson in 1967. In this article, we prove the following more general result: $τ(f( |Φ(A)|)) \le τ(f( |A|))$ for $A \in \mathfrak{A}$ and any increasing continuous function $f : [0, \infty) \to \mathbb{R}$ such that $f \circ \exp$ is convex on $\mathbb{R}$. Under the additional hypotheses that $A$ is invertible in $\mathscr{M}$ and $f \circ \exp$ is strictly convex, we have $τ(f( |Φ(A)|)) = τ(f( |A|))$ $\Longleftrightarrow Φ(A) = A$. As an application, we show that for $A \in \mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of $Φ(A)$, though such a conclusion does not hold in general for their spectra.

math.OA↗

arXiv2018

An operator inequality for range projections

By a result of Lundquist-Barrett, it follows that the rank of a positive semi-definite matrix is less than or equal to the sum of the ranks of its principal diagonal submatrices when written in block form. In this article, we take a general operator algebraic approach which provides insight as to why the above rank inequality resembles the Hadamard-Fischer determinant inequality in form, with multiplication replaced by addition. It also helps in identifying the necessary and sufficient conditions under which equality holds. Let $\mathscr{R}$ be a von Neumann algebra, and $Φ$ be a normal conditional expectation from $\mathscr{R}$ onto a von Neumann subalgebra $\mathscr{S}$ of $\mathscr{R}$. Let $\mathfrak{R}[T]$ denote the range projection of an operator $T$. For a positive operator $A$ in $\mathscr{R}$, we prove that $Φ(\mathfrak{R}[A]) \le \mathfrak{R}[Φ(A)]$ with equality if and only if $\mathfrak{R}[A] \in \mathscr{S}$.

math.OA↗

arXiv2018

On a class of determinant preserving maps for finite von Neumann algebras

Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $τ$ and let $Δ$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $ϕ$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$Δ(ϕ(A)+ϕ(B)) = Δ(A+B).$$ We show that any such map originates from a $τ$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either $*$-automorphism or $*$-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation $Δ(A + B) = Δ(A) + Δ(B)$ in the set of invertible positive operators in $\mathscr{R}$. To this end, we give a new proof of the inequality $$Δ(A+B) \ge Δ(A) + Δ(B),$$ using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible $B$ if and only if $A$ is a nonnegative scalar multiple of $B$.

math.OA↗

arXiv2017

The Hadamard Determinant Inequality - Extensions to Operators on a Hilbert Space

A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some `Fischer-type' inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction with the theory of operator monotone and operator convex functions on $[0,\infty)$. We place emphasis on documenting necessary and sufficient conditions for equality to hold.

math.OA↗