Questions tagged [operator-norms]

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Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
Mikhail Ostrovskii's user avatar
12 votes
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Does this matrix norm inequality have interesting application in other areas of mathematics?

In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices: Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric ...
Mostafa's user avatar
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Existence of more than two C*-norms on algebraic tensor product of C*-algebras

Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$. If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
Alcides Buss's user avatar
7 votes
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Minimizing the operator norm of a sum of matrices

Given an $m\times n$ real matrix $C$. Let $a_i\in\{-1,1\}$ and $b_i\in\{-1,1\}$. Consider $$q=\left\|\text{diag}(a_1,\ldots,a_m)C + C\text{diag}(b_1,\ldots,b_n)\right\|$$ where the norm is the ...
Wouter's user avatar
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Schur norm of weighted Cauchy matrix

The Schur norm of a matrix $A$ is defined to be $\|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$, where $\|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value. Let $a_1,\ldots, ...
user07001129's user avatar
5 votes
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Operator norm of a soft thresholded symmetric matrix

Let $A$ be a symmetric real-valued $n\times n$ matrix and let ${\left\|A\right\|_{2\rightarrow 2}} := \max_{\left\|u\right\|_{2}\leq 1} \left\|Au\right\|_{2}$ denotes its operator norm (largest ...
Student's user avatar
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Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = [G\hspace{1em}\...
dohmatob's user avatar
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390 views

An operator-norm version of Siegel's Lemma

Is there a kind of Siegel's Lemma saying that if $M$ is a ``small-height'' integer matrix, then there is a "small-height" vector $x$ with $\|Mx\|=\|M\|\|x\|$? (Here $\|Mx\|$ and $\|x\|$ denote the ...
Seva's user avatar
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Matrices with almost constant coefficient have a simple eigenvalue

As a by-product of a general result for bounded operators of a Banach space, I have the following: A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
Benoît Kloeckner's user avatar
4 votes
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Boundedness criterion for operators on mixed Lebesgue spaces

Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences ${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that ...
Philipp's user avatar
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Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
arriopolis's user avatar
3 votes
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An elementary proof of Davies' inequality

In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality. Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...
user07001129's user avatar
3 votes
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Araki's proof of simple connectedness of the restricted orthogonal group

I am trying to understand Araki's proof of the statement that the restricted orthogonal group of a Hilbert space with a unitary structure is simply connected. This proof starts on page 114 of these ...
Peter's user avatar
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Norm of a singular integral operator

Let $H$ be the characteristic function of $(0,+\infty)$ and let us define for $(x,y)\in \mathbb R^2$, $x\not=y$ $$ k(x,y)=\frac{H(x+y)}{iπ(x-y)}. $$ For $u\in C^1_c(\mathbb R)$, we define for $x\in \...
Bazin's user avatar
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Do Isometry Groups Tell Us How Difficult Norms are to Compute?

The question: Consider two norms N1 and N2 on the space of n-by-n complex matrices. N1 and N2 have the same isometry group and computing N1 is NP-HARD. Does it follow that computing N2 is NP-HARD as ...
Nathaniel Johnston's user avatar
2 votes
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155 views

which operators are "really truly positive"?

Let's say that an operator G on a Hilbert space $\mathcal{H}$ is "really truly positive" iff $\Vert\exp(-tG) \exp(-tG^*)\Vert_{op}<1$ for all $t>0$ How can we characterize the set of operators ...
Jackson Loper's user avatar
2 votes
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265 views

Maximum spectral norm of matrices with given anti-Hermitian part and Hermitian part's spectrum

Let $M\in M_n(\mathbb C)$ be a $n\times n$ matrix over the complex field. It can be written uniquely as $M=H+A$, where $H=H^*$ denotes its Hermitian part and $A=-A^*$ its anti-Hermitian part. Its ...
francesco999's user avatar
2 votes
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On design of a (preferrably unimodular) matrix

Assume each entry is in $\Bbb Z$. Say we want to solve $Ax=b$ where known $A$ is $n\times n$, unknown $x$ is $n\times1$ and $b$ is $n\times1$. The absolute value of minors of augmented matrix $[A|b]$...
Turbo's user avatar
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2 votes
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Rank of the smallest rank-reducing perturbation of a general matrix

Let $A$ be a general $m \times n$ matrix (not necessarily square, and not necessarily of full rank). Let $\| \cdot \|$ be a norm for $m \times n$ matrices that is induced from a norm for $m$ vectors ...
Joseph Grcar's user avatar
1 vote
0 answers
183 views

Schatten norm inequality

Let $A,B$ be two $n\times n$ matrices. Find a lower bound of the $p$-th Schatten norm $\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
volond's user avatar
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How to numerically compute the operator norm of an operator acting on a matrix algebra?

Let $M_n(C)$ denote the $n\times n$ matrices with complex entries acting on the Hilbert space $C^n$. As norm on $M_n(C)$ we take the operator norm, i.e. the largest eigenvalue of its absolute value. ...
Matthijs's user avatar
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102 views

Equivalence constants for induced matrix norms

Disclaimer: I asked this question beforehand on mathematics stack exchange, but I think it is better suited for this site Given two sets $P_i\in\mathbb{R}^s$, bounded, convex, with non-empty interior ...
tommsch's user avatar
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Row-wise conjugation of completely bounded map by group action

Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...
AlexE's user avatar
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specific sequence of matrices making a strange ratio of matrix norms diverging

For any $t>0$ define $d_t:=\operatorname{diag}j^t=\operatorname{diag}(1^t,2^t,\ldots)$. Now pick up such a $t>0$ and an arbitrary $\theta\in\big(0,\frac12\big)$. For every $k\in\mathbb{N}$ find ...
Krzysztof's user avatar
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Operator norm for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$

Suppose $C$ is a $n$ by $n$ real symmetric matrix, and $x\in R^n$. Is there an operator norm of $C$ for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$? If I decompose $C$ into $A'A = C^{-1}$, It seems ...
ahala's user avatar
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1 vote
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comparing norms of block-matrices

Let $k\in\mathbb{N}$ and let $d\in\mathcal{M}_k=\mathcal{B}(\ell_2^k)$ be a $k\times k$ diagonal matrix with strictly positive entries. Let now $x_1,\ldots,x_m\in\mathcal{M}_k\,\,(m\in\mathbb{N}$ - ...
Krzysztof's user avatar
  • 351
1 vote
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505 views

Epsilon-net of operator norm ball around Identity

Suppose I look at the set of matrices which are invertible and satisfy $$ \left\|A-Id\right\|_{op}<r $$ for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...
squattyroo's user avatar
1 vote
0 answers
393 views

Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality. $$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$ where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
Brian Ding's user avatar
1 vote
0 answers
438 views

Bound the expectation of trace norm of random Hermitian matrix

Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$. We would like to bound the following expectation on the trace norm $\mathbb{E}|\sum_{...
gondolf's user avatar
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1 vote
0 answers
104 views

expansion with respect to p-norms for p other than 2

Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$. Let $x \in {\...
Cristopher Moore's user avatar
1 vote
0 answers
277 views

Tensors with low spectral norm

Consider a tensor $T$ with six indices, $T_{(ii')(jj')(kk')}$, where each index goes from $1$ to $n$. We can think of $T$ as a linear map from $\mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$ ...
Michal Kotowski's user avatar
0 votes
0 answers
48 views

In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel

For a fixed distribution $p^{in}$ on the input space $ \mathbb{R}^{n_0}$, consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$. On this space, ...
Shuofeng Zhang's user avatar
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58 views

Explicit description for dual to operator space of Hilbert space

Let $H$ be a separable Hilbert space, and $B := \mathcal B(H)$ be the space of bounded operators on $H$. It is known that $B^\ast_\mathrm{strong} = B^\ast_\mathrm{weak}$ (see [Dunford, Schwartz, VI.1....
Vasily Ionin's user avatar