arXiv subjects

K. Mahesh Krishna

Publications and source records attributed to K. Mahesh Krishna.

At least 73 records · Page 4Linked to original sources

arXiv2022

Metric, Schauder and Operator-Valued Frames

Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \ It is proved that every separable metric space admits a metric $\mathcal{M}_d$-frame. Through Lipschitz-free Banach spaces it is showed that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. Several characterizations of metric frames are obtained. Stability results are also presented. Non linear multipliers are introduced and studied. This notion is connected with the notion of Lipschitz compact operators. Continuity properties of multipliers are discussed. For a subclass of approximated Schauder frames for Banach spaces, characterization result is derived using standard Schauder basis for standard sequence spaces. Duals of a subclass of approximate Schauder frames are completely described. Similarity of this class is characterized and interpolation result is derived using orthogonality. A dilation result is obtained. A new identity is derived for Banach spaces which admit a homogeneous semi-inner product. Some stability results are obtained for this class. A generalization of operator-valued frames for Hilbert spaces are introduced which unifies all the known generalizations of frames for Hilbert spaces. This notion has been studied in depth by imposing factorization property of the frame operator. Its duality, similarity and orthogonality are addressed. Connections between this notion and unitary representations of groups and group-like unitary systems are derived. Paley-Wiener theorem for this class are derived.

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arXiv2022

The $(abc,pqr)$-problem for Approximate Schauder Frames for Banach Spaces

Motivated from the complete solution of important $abc$-problem for Gabor system for the Hilbert space $\mathcal{L}^2(\mathbb{R})$ by Dai and Sun [\textit{Memoirs of Amer. Math. Soc., 2016}] and from the existential result of approximate Schauder frames for $\mathcal{L}^p(\mathbb{R})$ using translation operators on $\mathcal{L}^p(\mathbb{R})$ by Freeman, Odell, Schlumprecht, and Zsak [\textit{Israel. J. Math, 2014}], we formulate $(abc, pqr)$-problem for approximate Schauder frames for Banach spaces $\mathcal{L}^p(\mathbb{R})$, $1<p<\infty$.

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arXiv2022

Operator-Valued p-Approximate Schauder Frames

We give an operator-algebraic treatment of theory of p-approximate Schuader frames which includes the theory of operator-valued frames by Kaftal, Larson, and Zhang [\textit{Trans. AMS., 2009}], G-frames by Sun [JMAA, 2006], factorable weak operator-valued frames by Krishna and Johnson [\textit{Annals of FA, 2022}] and p-approximate Schauder frames by Krishna and Johnson [\textit{J. Pseudo-Differ. Oper. Appl, 2021}] as particular cases. We show that a sufficiently rich theory can be developed even for Banach spaces. We achieve this by defining various concepts and characterizations in Banach spaces. These include duality, approximate duality, equivalence, orthogonality and stability.

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arXiv2022

Paulsen and Projection Problems for Banach Spaces

Based on the two decades old celebrated Paulsen problem and its solutions for Hilbert spaces by Kwok, Lau, Lee, Ramachandran, Hamilton, and Moitra, we formulate Paulsen problem for Banach spaces. We also formulate projection problem for Banach spaces.

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arXiv2022

Discrete and Continuous Welch Bounds for Banach Spaces with Applications

Let $\{τ_j\}_{j=1}^n$ be a collection in a finite dimensional Banach space $\mathcal{X}$ of dimension $d$ and $\{f_j\}_{j=1}^n$ be a collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) such that $f_j(τ_j) =1$, $\forall 1\leq j\leq n$. Let $n\geq d$ and $\text{Sym}^m(\mathcal{X})$ be the Banach space of symmetric m-tensors. If the operator $ \text{Sym}^m(\mathcal{X})\ni x \mapsto \sum_{j=1}^nf_j^{\otimes m}(x)τ_j ^{\otimes m}\in\text{Sym}^m(\mathcal{X})$ is diagonalizable and its eigenvalues are all non negative, then we prove that \begin{align}\label{WELCHBANACHABSTRACT} \max _{1\leq j,k \leq n, j\neq k}|f_j(τ_k)|^{2m}\geq \max _{1\leq j,k \leq n, j\neq k}|f_j(τ_k)f_k(τ_j)|^m \geq\frac{1}{n-1}\left[\frac{n}{d+m-1\choose m}-1\right], \quad \forall m \in \mathbb{N}. \end{align} When $ \mathcal{X}=\mathcal{H}$ is a Hilbert space, and $f_j$ is defined by $f_j: \mathcal{H}\ni h \mapsto \langle h, τ_j \rangle \in \mathbb{K}$ (where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$), $\forall 1 \leq j \leq n$, then Inequality (1) reduces to Welch bounds. Thus Inequality (1) improves 48 years old result obtained by Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We also prove the following continuous version of Inequality (1) under certain conditions for measure spaces: \begin{align}\label{CONTINUOUSWELCHBANACHABSTRACT} \sup _{α, β\in Ω, α\neq β}|f_α(τ_β) |^{2m}\geq \sup _{α, β\in Ω, α\neq β}|f_α(τ_β)f_β(τ_α) |^{m}\geq \frac{1}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}\left[\frac{ μ(Ω)^2}{d+m-1 \choose m}-(μ\timesμ)(Δ)\right]. \end{align}

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arXiv2022

Modular Welch Bounds with Applications

We prove the following two results. \begin{enumerate} \item Let $\mathcal{A}$ be a unital commutative C*-algebra and $\mathcal{A}^d$ be the standard Hilbert C*-module over $\mathcal{A}$. Let $n\geq d$. If $\{τ_j\}_{j=1}^n$ is any collection of vectors in $\mathcal{A}^d$ such that $\langle τ_j, τ_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k \leq n, j\neq k}\|\langle τ_j, τ_k\rangle ||^{2m}\geq \frac{1}{n-1}\left[\frac{n}{d+m-1\choose m}-1\right], \quad \forall m \in \mathbb{N}. \end{align*} \item Let $\mathcal{A}$ be a $σ$-finite commutative W*-algebra or a commutative AW*-algebra and $\mathcal{E}$ be a rank d Hilbert C*-module over $\mathcal{A}$. Let $n\geq d$. If $\{τ_j\}_{j=1}^n$ is any collection of vectors in $\mathcal{E}$ such that $\langle τ_j, τ_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k \leq n, j\neq k}\|\langle τ_j, τ_k\rangle ||^{2m}\geq \frac{1}{n-1}\left[\frac{n}{d+m-1\choose m}-1\right], \quad \forall m \in \mathbb{N}. \end{align*} \end{enumerate} Results (1) and (2) reduce to the famous result of Welch [\textit{IEEE Transactions on Information Theory, 1974}] obtained 48 years ago. We introduce the notions of modular frame potential, modular equiangular frames and modular Grassmannian frames. We formulate Zauner's conjecture for Hilbert C*-modules.

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arXiv2022

Feichtinger Conjectures, $R_\varepsilon$-Conjectures and Weaver's Conjectures for Banach spaces

Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality for frames for Hilbert spaces and scaling problem for Hilbert space frames. Based on Kothe-Lorch theorem for Riesz bases for Hilbert spaces, we formulate a problem for p-approximate Riesz bases for Banach spaces. We formulate dynamical sampling problem for p-approximate Schauder frames for Banach spaces. We ask phase retrieval problem and norm retrieval problem for p-approximate Schauder frames for Banach spaces. We also formulate discretization problem for continuous p-approximate Schauder frames.

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arXiv2021

Approximately Dual p-Approximate Schauder Frames

Difficulty in the construction of dual frames for a given Hilbert space led to the introduction of approximately dual frames in Hilbert spaces by Christensen and Laugesen. It becomes even more difficult in Banach spaces to construct duals. For this purpose, we introduce approximately dual frames for a class of approximate Schauder frames for Banach spaces and develop basic theory. Approximate duals for this subclass is completely characterized and its perturbation is also studied.

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arXiv2021

Continuous Welch bounds with Applications

Let $(Ω, μ)$ be a measure space and $\{τ_α\}_{α\in Ω}$ be a normalized continuous Bessel family for a finite dimensional Hilbert space $\mathcal{H}$ of dimension $d$. If the diagonal $Δ\coloneqq \{(α, α):α\in Ω\}$ is measurable in the measure space $Ω\times Ω$, then we show that \begin{align*} \sup _{α, β\in Ω, α\neq β}|\langle τ_α, τ_β\rangle |^{2m}\geq \frac{1}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}\left[\frac{ μ(Ω)^2}{d+m-1 \choose m}-(μ\timesμ)(Δ)\right], \quad \forall m \in \mathbb{N}. \end{align*} This improves 47 years old celebrated result of Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We introduce the notions of continuous cross correlation and frame potential of Bessel family and give applications of continuous Welch bounds to these concepts. We also introduce the notion of continuous Grassmannian frames.

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arXiv2021

Improving Casazza-Kalton-Christensen-van Eijndhoven Perturbation with Applications

Let $ \mathcal{X}$, $ \mathcal{Y}$ be Banach spaces and $S:\mathcal{X} \to \mathcal{Y} $ be an invertible Lipschitz map. Let $ T : \mathcal{X}\rightarrow \mathcal{Y}$ be a map and there exist $ λ_1,λ_2 \in \left [0, 1 \right )$ such that \begin{align*} \|Tx-Ty-(Sx-Sy)\|\leqλ_1\|Sx-Sy\|+λ_2\|Tx-Ty\|,\quad \forall x,y \in \mathcal{X}. \end{align*} Then we prove that $T$ is an invertible Lipschitz map. This improves 25 years old Casazza-Kalton-Christensen-van Eijndhoven perturbation. It also improves 28 years old Soderlind-Campanato perturbation and 2 years old Barbagallo-Ernst-Thera perturbation. We give applications to the theory of metric frames. The notion of Lipschitz atomic decomposition for Banach spaces is also introduced.

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arXiv2021

C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture

Striking result of Vyb\'ıral [\textit{Adv. Math.} 2020] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vyb\'ıral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vyb\'ıral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate Pólya-Szegő-Rudin question for the C*-algebraic Schur product of positive matrices.

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arXiv2021

Multipliers for operator-valued Bessel sequences, generalized Hilbert-Schmidt and trace classes

Let $\{λ_n\}_n \in \ell^\infty(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}λ_n (x_n\otimes \bar{y_n})$, where $\{x_n\}_n$, $\{y_n\}_n$ are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \cite{BALAZS3} generalized this by replacing $\{x_n\}_n$ and $\{y_n\}_n$ by Bessel sequences. In this paper, we generalize this by studying the operators of the form $\sum_{n=1}^{\infty}λ_n (A^*_nx_n\otimes \bar{B^*_ny_n})$, where $\{A_n\}_n$ and $\{B_n\}_n$ are operator-valued Bessel sequences and $\{x_n\}_n$, $\{y_n\}_n$ are sequences in the Hilbert space such that $\{\|x_n\|\|y_n\|\}_n \in \ell^\infty(\mathbb{N})$. We next generalize the classes of Hilbert-Schmidt and trace class operators.

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arXiv2021

Dilations of Linear Maps on Vector Spaces

We continue the study dilation of linear maps on vector spaces introduced by Bhat, De, and Rakshit. This notion is a variant of vector space dilation introduced by Han, Larson, Liu, and Liu. We derive vector space versions of Wold decomposition, Halmos dilation, N-dilation, inter-twining lifting theorem and a variant of Ando dilation. It is noted further that unlike a kind of uniqueness of Halmos dilation of strict contractions on Hilbert spaces, vector space version of Halmos dilation can not be characterized.

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arXiv2021

Commutators Close to the Identity in Unital C*-Algebras

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ be the C*-algebra of all bounded linear operators on $\mathcal{H}$, equipped with the operator-norm. By improving the Brown-Pearcy construction, Terence Tao in 2018, extended the result of Popa [1981] which reads as : For each $0<\varepsilon\leq 1/2$, there exist $D,X \in \mathcal{B}(\mathcal{H})$ with $\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\|\leq \varepsilon$ such that $\|D\|\|X\|=O\left(\log^5\frac{1}{\varepsilon}\right)$, where $[D,X]:= DX-XD$. In this paper, we show that Tao's result still holds for certain class of unital C*-algebras which include $\mathcal{B}(\mathcal{H})$ as well as the Cuntz algebra $\mathcal{O}_2$.

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arXiv2021

Expansion of approximate Bessel sequences to approximate Schauder frames for Banach spaces

It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.

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arXiv2021

New Identity on Parseval p-Approximate Schauder Frames and Applications

A very useful identity for Parseval frames for Hilbert spaces was obtained by Balan, Casazza, Edidin, and Kutyniok. In this paper, we obtain a similar identity for Parseval p-approximate Schauder frames for Banach spaces which admits a homogeneous semi-inner product in the sense of Lumer-Giles.

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arXiv2020

Perturbation of p-approximate Schauder frames for separable Banach spaces

Paley-Wiener theorem for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach spaces. We show that our results give Paley-Wiener theorem for frames for Hilbert spaces.

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arXiv2020

Dilation theorem for p-approximate Schauder frames for separable Banach spaces

Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $\mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $\mathcal{H}_1$ which contains $\mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.

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