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K. Mahesh Krishna

Publications and source records attributed to K. Mahesh Krishna.

At least 55 records · Page 3Linked to original sources
arXiv2023

Functional Ghobber-Jaming Uncertainty Principle

Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{ω_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} o(M)^\frac{1}{q}o(N)^\frac{1}{p}< \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(τ_j) |}, \end{align*} where $q$ is the conjugate index of $p$. Then for all $x \in \mathcal{X}$, we show that \begin{align}\label{FGJU} (1) \quad \quad \quad \quad \|x\|\leq \left(1+\frac{1}{1-o(M)^\frac{1}{q}o(N)^\frac{1}{p}\displaystyle\max_{1\leq j,k\leq n}|g_k(τ_j)|}\right)\left[\left(\sum_{j\in M^c}|f_j(x)|^p\right)^\frac{1}{p}+\left(\sum_{k\in N^c}|g_k(x) |^p\right)^\frac{1}{p}\right]. \end{align} We call Inequality (1) as \textbf{Functional Ghobber-Jaming Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}.

math.FA↗
arXiv2023

Group-Frames for Banach Spaces

In the literature, frames generated by unitary representations of groups (known as group-frames) are studied only for Hilbert spaces. We make first study of frames for Banach spaces generated by isometric invertible representations of discrete groups on Banach spaces. These frames are characterized using left regular, right regular, Gram-matrices and group-matrices on classical sequence spaces. A sufficiently large collection of functional-vector pairs using the double commutant of the representation is identified which generate group-frames for Banach spaces. Subsequently, we study Schauder frames generated by time-frequency shift operators on finite dimensional Banach spaces. We derive Moyal formula, fundamental identity of Gabor analysis, Wexler-Raz criterion and Ron-Shen duality in functional form.

math.FA↗
arXiv2023

Absolutely Summing Morphisms between Hilbert C*-Modules and Modular Pietsch Factorization Problem

Motivated from the theory of Hilbert-Schmidt morphisms between Hilbert C*-modules over commutative C*-algebras by Stern and van Suijlekom \textit{[J. Funct. Anal., 2021]}, we introduce the notion of p-absolutely summing morphisms between Hilbert C*-modules over commutative C*-algebras. We show that an adjointable morphism between Hilbert C*-modules over monotone closed commutative C*-algebra is 2-absolutely summing if and only if it is Hilbert-Schmidt. We formulate version of Pietsch factorization problem for p-absolutely summing morphisms and solve partially

math.FA↗
arXiv2023

Localized Bounded Below Approximate Schauder Frames are Finite Unions of Approximate Riesz Sequences

Based on the truth of Feichtinger conjecture by Marcus, Spielman and Srivastava \textit{[Ann. of Math. (2), 2015]} and from the localized version by Gröchenig \textit{[Adv. Comput. Math., 2003]}, we introduce the notion of localization of approximate Schauder frames (ASFs) and approximate Riesz sequences (ARSs). We show that localized bounded below ASFs are finite unions of ARSs.

math.FA↗
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.

math.FA↗
arXiv2022

Algebraic Andô Dilation

We solve the Andô dilation problem for linear maps on vector space asked by Krishna and Johnson in \textit{[Oper. Matrices, 2022]}. More precisely, we show that any commuting linear maps on vector space can be dilated to commuting injective linear maps.

math.FA↗
arXiv2022

Indefinite Halmos, Egervary and Sz.-Nagy Dilations

Let $\mathcal{M}$ be an indefinite inner product module over a *-ring of characteristic 2. We show that every self-adjoint operator on $\mathcal{M}$ admits Halmos, Egervary and Sz.-Nagy dilations.

math.RA↗
arXiv2022

Continuous Non-Archimedean and p-adic Welch Bounds

We prove the continuous non-Archimedean (resp. p-adic) Banach space and Hilbert space versions of non-Archimedean (resp. p-adic) Welch bounds proved by M. Krishna. We formulate continuous non-Archimedean and p-adic functional Zauner conjectures.

math.NT↗
arXiv2022

Non-Archimedean and p-adic Functional Welch Bounds

We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. 1. Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2$ for all $ λ_j \in \mathbb{K}, 1\leq j \leq n$, for all $n \in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) satisfying $f_j(τ_j) =1$ for all $1\leq j \leq n$ and the operator $S_{f, τ} : \text{Sym}^m(\mathcal{X})\ni x \mapsto \sum_{j=1}^nf_j^{\otimes m}(x)τ_j^{\otimes m} \in \text{Sym}^m(\mathcal{X})$, is diagonalizable, then \begin{align} \text{(Non-Archimedean Functional Welch Bounds)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(τ_k)f_k(τ_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} 2. For a prime $p$, let $\mathbb{Q}_p$ be the p-adic number field. Let $\mathcal{X}$ be a $d$-dimensional p-adic Banach space over $\mathbb{Q}_p$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) satisfying $f_j(τ_j) =1$ for all $1\leq j \leq n$ and there exists $b \in \mathbb{Q}_p$ such that $ \sum_{j=1}^{n}f_j^{\otimes m}(x) τ_j^{\otimes m} =bx$ for all $ x \in \text{Sym}^m(\mathcal{X}),$ then \begin{align} \text{(p-adic Functional Welch Bounds)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(τ_k)f_k(τ_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We formulate non-Archimedean functional and p-adic functional Zauner conjectures.

math.NT↗
arXiv2022

p-adic Welch Bounds and p-adic Zauner Conjecture

Let $p$ be a prime. For $d\in \mathbb{N}$, let $\mathbb{Q}_p^d$ be the standard $d$-dimensional p-adic Hilbert space. Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{Q}_p^d)$ be the p-adic Hilbert space of symmetric m-tensors. We prove the following result. Let $\{τ_j\}_{j=1}^n$ be a collection in $\mathbb{Q}_p^d$ satisfying (i) $\langle τ_j, τ_j\rangle =1$ for all $1\leq j \leq n$ and (ii) there exists $b \in \mathbb{Q}_p$ satisfying $ \sum_{j=1}^{n}\langle x, τ_j\rangle τ_j =bx$ for all $ x \in \mathbb{Q}^d_p.$ Then \begin{align} (1) \quad \quad \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle τ_j, τ_k\rangle|^{2m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We call Inequality (1) as the p-adic version of Welch bounds obtained by Welch [\textit{IEEE Transactions on Information Theory, 1974}]. Inequality (1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate p-adic Zauner conjecture.

math.NT↗
arXiv2022

Non-Archimedean Welch Bounds and Non-Archimedean Zauner Conjecture

Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying \begin{align*} \left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2, \quad \forall λ_j \in \mathbb{K}, 1\leq j \leq n, \forall n \in \mathbb{N}. \end{align*} For $d\in \mathbb{N}$, let $\mathbb{K}^d$ be the standard $d$-dimensional non-Archimedean Hilbert space. Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{K}^d)$ be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If $\{τ_j\}_{j=1}^n$ is a collection in $\mathbb{K}^d$ satisfying $\langle τ_j, τ_j\rangle =1$ for all $1\leq j \leq n$ and the operator $\text{Sym}^m(\mathbb{K}^d)\ni x \mapsto \sum_{j=1}^n\langle x, τ_j^{\otimes m}\rangle τ_j^{\otimes m} \in \text{Sym}^m(\mathbb{K}^d)$ is diagonalizable, then \begin{align} (1) \quad \quad \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle τ_j, τ_k\rangle|^{2m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We call Inequality (1) as the non-Archimedean version of Welch bounds obtained by Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We formulate non-Archimedean Zauner conjecture.

cs.IT↗
arXiv2022

Modular Paulsen Problem and Modular Projection Problem

Based on the solution of \textbf{Paulsen Problem} by Kwok, Lau, Lee, and Ramachandran [\textit{STOC'18-Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018}] and independently by Hamilton, and Moitra [\textit{Isr. J. Math., 2021}] we study Paulsen Problem and Projection Problem in the context of Hilbert C*-modules. We show that for commutative C*-algebras, if Modular Paulsen Problem has a solution, then Modular Projection Problem also has a solution. We formulate the problem of operator scaling for matrices over C*-algebras.

math.FA↗
arXiv2022

C*-algebraic Casas-Alvero Conjecture

Based on Casas-Alvero conjecture \textit{[J. Algebra, 2001]} we formulate the following conjecture.\\ \textbf{C*-algebraic Casas-Alvero Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra, $n\in \mathbb{N}$ and let $P(z) = (z-a_1)(z-a_2)\cdots (z-a_n)$ be a polynomial over $\mathcal{A}$ with $a_1, a_2, \dots, a_n \in \mathcal{A}$. If $P$ shares a common zero with each of its (first) $n-1$ derivatives, then it is $n^\text{th}$ power of a linear monic C*-algebraic polynomial.}\\ We show that C*-algebraic Casas-Alvero Conjecture holds for C*-algebraic polynomials of degree 2.

math.FA↗
arXiv2022

C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture

Based on Smale mean value conjecture \textit{[Bull. Amer. Math. Soc., 1981]} and Dubinin-Sugawa dual mean value conjecture \textit{[Proc. Japan Acad. Ser. A Math. Sci., 2009]} we formulate the following conjectures. \textbf{C*-algebraic Smale Mean Value Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\geq 2$ over $\mathcal{A}$, $a_1, \dots, a_n \in \mathcal{A}$. If $z\in\mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that \begin{align*} \frac{\|P(z)-P(w)\|}{\|z-w\|}\leq 1 \|P'(z)\| \end{align*} or \begin{align*} \frac{\|P(z)-P(w)\|}{\|z-w\|}\leq \frac{n-1}{n} \|P'(z)\|=\frac{\operatorname{deg} (P)-1}{\operatorname{deg} (P)} \|P'(z)\|. \end{align*}} \textbf{C*-algebraic Dubinin-Sugawa Dual Mean Value Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\geq 2$ over $\mathcal{A}$, $a_1, \dots, a_n \in \mathcal{A}$. If $z\in \mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that \begin{align*} \frac{\|P'(z)\|}{\operatorname{deg} (P)} =\frac{\|P'(z)\|}{n} \leq \frac{\|P(z)-P(w)\|}{\|z-w\|}. \end{align*}} We show that (even a strong form of) C*-algebraic Smale mean value conjecture and C*-algebraic Dubinin-Sugawa dual mean value conjecture hold for degree 2 C*-algebraic polynomials over commutative C*-algebras.

math.FA↗
arXiv2022

C*-algebraic Schoenberg Conjecture

Based on Schoenberg conjecture \textit{[Amer. Math. Monthly., 1986]}/Malamud-Pereira theorem \textit{[J. Math. Anal. Appl, 2003]}, \textit{[Trans. Amer. Math. Soc., 2005]} we formulate the following conjecture which we call C*-algebraic Schoenberg Conjecture.\\ \textbf{ C*-algebraic Schoenberg Conjecture : Let $\mathcal{A}$ be a C*-algebra. Let $d\in \mathbb{N}\setminus\{1\}$, $P(z)= (z-a_1)(z-a_2)\cdots (z-a_d)$ be a polynomial over $\mathcal{A}$ with $a_1, a_2, \dots, a_d \in \mathcal{A} $. If $P'$ can be written as $P'(z)= d(z-b_1)(z-b_2)\cdots (z-b_{d-1})$ on $\mathcal{A}$ with $b_1, b_2, \dots, b_{d-1} \in \mathcal{A} $, then \begin{align*} \sum_{k=1}^{d-1}b_kb_k^*\leq \frac{1}{d^2}\left[\sum_{j=1}^{d}a_j\right]\left[\sum_{j=1}^{d}a_j\right]^*+ \frac{d-2}{d}\sum_{j=1}^{d}a_ja_j^* \end{align*} and \begin{align*} \sum_{k=1}^{d-1}b_k^*b_k\leq \frac{1}{d^2}\left[\sum_{j=1}^{d}a_j\right]^*\left[\sum_{j=1}^{d}a_j\right]+ \frac{d-2}{d}\sum_{j=1}^{d}a_j^*a_j. \end{align*}} We show that C*-algebraic Schoenberg conjecture holds for degree 2 C*-algebraic polynomials over C*-algebras.

math.OA↗
arXiv2022

C*-algebraic Gauss-Lucas Theorem and C*-algebraic Sendov's Conjecture

Using a result of Robertson \textit{[Proc. Edinburgh Math. Soc. (2), 1976]}, we introduce a notion of differentiation of maps on certain classes of unital commutative C*-algebras. We then derive C*-algebraic Gauss-Lucas theorem and formulate C*-algebraic Sendov's conjecture. We verify C*-algebraic Sendov's conjecture for polynomials of degree 2.

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