Questions tagged [operator-norms]
The operator-norms tag has no usage guidance.
117
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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 ...
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Hölder's inequality for matrices
I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...
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votes
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answer
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Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector?
I'd like to state explicitly a problem which was somehow hidden in my three-week-old post:
Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal M}_{m\...
- 22.8k
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votes
1
answer
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A matrix completion problem
In their paper, Corners of normal matrices, Rajendra Bhatia and Man-Duen Choi asked the following question:
Given a matrix pair $(B,C)$ where $B,C∈M_n$, does there exist matrices $A,D ∈ M_n$ such ...
- 3,679
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$\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$
I'm interested in
$$\sup_{x, y} \left\| A x + B y\right\|_2$$ subject to
$$\left\|x\right\|_2 = \left\|y\right\|_2 = 1$$
where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of ...
- 1,617
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2
answers
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Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$
The setup is as in this question:
Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}...
- 13.2k
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votes
1
answer
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Operator norms of circulant matrices
The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
\mbox{...
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Matrix trace & norm [closed]
For any nonnegative semidefinite matrix $A$ and any matrix $B$, we have
$$\mbox{tr} (AB) \le \mbox{tr} (A) \, \|B\|$$
where $\mbox{tr}(\cdot)$ is the trace and $\|\cdot\|$ is the operator norm. How ...
- 157
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Norm of $n$-linear symmetric forms
Let $B$ be a symmetric bilinear form over a Euclidean space $E$. Say that $|B(v,v)|\le c\|v\|^2$ for every $v\in E$, for some $c\ge0$. Then
$$4B(v,w)=B(v+w)+B(v-w)$$
yields $2|B(v,w)|\le c(\|v\|^2+\|w\...
- 51.6k
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votes
2
answers
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which norms can be realized as operator norms?
Assume $(V,∥∥_V),(W,∥∥_W)$ are both finite dimensional normed spaces. We have the induced operator norm on ${\rm Hom}(V,W)$.
It turns out that the operator norm is induced by an inner product iff ...
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What is the "correct" generalization of operator norms for nonlinear operators?
I have been recently wondering what is a (or even the) "correct" generalization of the notion of an operator norm to nonlinear operators?
Please excuse the naivete of my question; if you think that ...
- 28.4k
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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 ...
- 4,454
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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 $(A,B)$...
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Bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?
$\DeclareMathOperator\Tr{Tr}$Let $A_i$ with $i=1,\dotsc,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $\Tr(p)=1$. Let $0< a_i<1$ and $\...
- 103
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Continuity of the product map
Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...
- 4,695
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votes
2
answers
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Equivalence between complex and real operator norms
Consider a real $m\times n$ matrix $A$ and the $p$-norms in $\mathbb{C}^n$ and $\mathbb{C}^m: \|x\|=\left(\sum|x_i|^p\right)^{1/p}$.
One defines the real $p$-norm of $A$ as $\|A\|=\sup\frac{\|Ax\|}{\|...
- 1,638
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votes
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answer
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Regular $p$-norm of a matrix
Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as ...
- 11.6k
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Continuously varying norms
Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d =...
- 1,093
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What is the Lipschitz constant of the differential of the matrix exponential $\mathfrak{so}(3)\to \mathrm{SO}(3)$
I'm interested in numerical methods on $\mathrm{SO}(3)$ manifold, and working on a particular problem using the exponential coordinates:
$$
R(u) := \exp(u_\times)
$$
with $u\in \mathbb{R}^3$ and where ...
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answers
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When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?
I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive ...
8
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1
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Equivalent Norms for the Dual of Sobolev / Bessel Spaces
Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb ...
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The norm of tensor product operator on Lp spaces
Let $X, Y$ be two $\sigma$-finite measure spaces and $p,q\in [1,\infty]$. Let $T_1, T_2:L^p(X)\rightarrow L^q(Y)$ be two bounded linear operators. Then one can define a linear operator $$T_1\otimes ...
- 71
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1
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Norm of tridiagonal operator
Recently, I needed to estimate the operator norm of the tridiagonal operator, but I am sure answers much more refined than my simple observations must be known.
Let $T$ be the linear operator that ...
- 28.4k
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1
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419
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Completely bounded norm for unital maps with completely positive sections
Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\...
- 1,394
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votes
2
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Characterizing when matrices are 'dissipative'
An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...
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Operator norm and spectrum
I am wondering about when an operator norm coincides with the maximum eigenvalue of an operator and there is one particular aspect that confuses me quite a lot.
Let's say we have a symmetric positive ...
- 71
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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 ...
- 201
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votes
3
answers
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Norm estimation of identity plus two non-commuting self-adjoint operators
This is a problem that I have been stuck for a few months.
Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that
$$\|(I+...
- 483
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2
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$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$
Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le \...
- 22.8k
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How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
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1
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Inequality between nuclear norm and operator norm for positive definite matrices
I will use $\|\|_*$ to denote the nuclear norm (sum of singular values) and $\|\|_2$ to denote the operator norm / matrix 2-norm (largest singular value).
Consider two positive definite $n \times n$ ...
- 61
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Absolutely 2-summable operator on a Hilbert space
An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that
$$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \right)^{1/2} ...
- 9,591
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1
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How can I see the "missing" Poisson center when the rank of the Poisson structure drops?
Recall that a Poisson algebra is a commutative algebra $A$ along with a bracket $\lbrace,\rbrace: A^{\otimes 2} \to A$ which is a Lie bracket and which is also a derivation in each variable. The ...
- 53.1k
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votes
1
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Recover approximate monotonicity of induced norms
Let $A$ some square matrix with real entries.
Take any norm $\|\cdot\|$ consistent with a vector norm.
Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$.
Moreover, ...
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Are Hilbert-Schmidt operators on separable Hilbert spaces "Hilbert Schmidt" on the space of Hilbert Schmidt Operators?
Let's consider a separable Hilbert space $(\mathcal H, \langle\cdot, \cdot\rangle_{\mathcal H})$ with Norm $||\cdot||_{\mathcal H} := \langle\cdot, \cdot\rangle^{1/2}_{\mathcal H},$ orthonomal basis $(...
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What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function?
The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup_{\delta>0,\left|y-x\right|<\delta} \text{...
- 1,973
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An extreme point of the ball of the space of compact operators
It is very easy to see that the unit ball of $c_0$ has no extreme points. I was trying to spot any extreme points in the unit ball of the space of compact operators on a Hilbert space (a non-...
- 51
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Euclidean norms of matrices
Let us consider the euclidean norm on $\mathbf{R}^2$. After some computations, I have obtained the following expression for the associated operator norm on 2 by 2 matrices.
$$
\left\lVert\pmatrix{a&...
- 18.5k
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1
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Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 175
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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, ...
- 81
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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 ...
- 159
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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}\...
- 6,726
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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 ...
- 22.8k
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2
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Operator norm versus Hlawka inequality
Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality
$${\bf(H)}\qquad\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\...
- 51.6k
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Operator norm of a masked SDP matrix
Let $\Sigma$ be a $d\times d$ semi-definite positive matrix (SDP). Let $I\subset\{1,\ldots, d\}\times \{1, \ldots, d\}$ be a symmetric subset of indices (i.e. if $(p,q)\in I$ then $(q,p)\in I$). We ...
4
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1
answer
197
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Retractions for completely positive unital maps, with particularly nice norms
Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \...
- 1,394
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CB-norm of mappings from a matrix space
The following result of Roger Smith is well known to operator algebraists:
$$\| \phi: E \rightarrow M_n\|_{cb} =\| \phi^{(n)} \otimes id_{M_n}: E \otimes_{min}M_n \rightarrow M_n \otimes M_n\|,$$
...
- 769
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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 $c$,...
- 14.2k
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votes
0
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185
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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
...
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trace norm inequality for positive matrices
If $A, B$ are positive $n \times n$ complex matrices, $n$ some integer, then obviously \begin{equation*} \|ABA\|_\text{tr} = tr(ABA) = tr(A^2 B). \end{equation*}
But can we say there is a constant $...
