Questions tagged [operator-norms]

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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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0 votes
1 answer
218 views

Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform

It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
Fei Cao's user avatar
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12 votes
0 answers
130 views

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
3 votes
2 answers
1k views

Is the matrix induced L1-norm greater than the induced L2-norm?

For $A \in \mathbb R^{m \times n}$ and the induced norms: $$ \| A \|_1 = \max_{x \ne 0} \frac{\|Ax\|_1}{\|x\|_1} $$ $$ \| A \|_2 = \max_{x \ne 0} \frac{\|Ax\|_2}{\|x\|_2} $$ ... where: $$ \|x\|_1 = \...
DrunkCoder's user avatar
1 vote
0 answers
186 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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1 vote
0 answers
199 views

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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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
3 votes
0 answers
281 views

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
0 votes
1 answer
259 views

Application of the Frechet derivative [closed]

$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that: \...
Julian Bejarano's user avatar
1 vote
0 answers
103 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
  • 119
5 votes
1 answer
364 views

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&...
coudy's user avatar
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1 vote
0 answers
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How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]

Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$. I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$. My belief is that this is true is motivated by empirical ...
Msc Splinter's user avatar
3 votes
1 answer
490 views

Trace norm of operators obtained by restricting the matrix of a trace class operator

Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...
NewB's user avatar
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1 vote
0 answers
54 views

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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9 votes
1 answer
287 views

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 =...
Jakob Werner's user avatar
  • 1,093
4 votes
1 answer
198 views

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 ...
guillaume lecue's user avatar
1 vote
1 answer
173 views

Some estimates on tensor norms

Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
A beginner mathmatician's user avatar
2 votes
1 answer
575 views

Confusing definition of homogeneous Sobolev norm of order -1

Let $\Omega \subset \mathbb{R}^{d}$ and $\|.\|$ is the standard euclidean $2$-norm. I came across a definition of $\dot{H}^{-1}(\Omega)$ which is a bit confusing. In [1] authors define the following ...
Titouan Vayer's user avatar
1 vote
1 answer
181 views

Example when Kantorovich condition would not hold

Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator $$ (T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+. $$ Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
user124297's user avatar
1 vote
1 answer
130 views

Embedding Hermitian forms into Hilbert spaces

Let $H$ be a nondegenerate, not positive definite, Hermitian form on a complex vector space $V$ such that $$|H(x,y)|\le u(x)u(y)\tag{B}$$ for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ ...
Arnold Neumaier's user avatar
7 votes
1 answer
2k views

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 ...
Peppermint's user avatar
9 votes
1 answer
530 views

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 ...
Maciej's user avatar
  • 111
6 votes
1 answer
275 views

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, ...
ippiki-ookami's user avatar
3 votes
0 answers
132 views

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
1 answer
114 views

Flatness directions of the operator norm

It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ...
Denis Serre's user avatar
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2 votes
0 answers
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
0 votes
1 answer
70 views

Equality between two norms on $\mathcal{L}(E)^n$

Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ the algebra of all bounded linear operators on $E$. On $\mathcal{L}(E)^n$, we have two equivalent norms: \begin{eqnarray*} N_1({\bf A}) &=&...
Student's user avatar
  • 1,154
1 vote
0 answers
327 views

Is the square of operator norm smooth? [closed]

Is $A \mapsto \|A\|_{op}^2$ a smooth function? Specifically, is the space of matrices equipped with operator norm a 2-smooth Banach space?
Arun's user avatar
  • 51
7 votes
2 answers
1k views

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 ...
Chris's user avatar
  • 71
5 votes
0 answers
89 views

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
0 votes
1 answer
174 views

Does $\{\left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right||\varphi\in\{\psi\}^{\perp}\}$ split $\mathfrak{S}_1$?

Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}...
user avatar
3 votes
2 answers
371 views

Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different

In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
worldreporter's user avatar
1 vote
1 answer
490 views

Relationship between $2 \to 2$ norm and $\infty \to 2$ norm [closed]

I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is. E.g., the trivial result is that for matrix $A$ with ...
Alex Wenxin Xu's user avatar
1 vote
1 answer
611 views

Minimal value of matrix norm induced by a norm

Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by $$ \| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}} $$ where the matrix $A$ is interpreted as an ...
Jonas Adler's user avatar
1 vote
1 answer
1k views

Inequality for the operator norm of a product of matrices

I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that $$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
BGJ's user avatar
  • 439
5 votes
0 answers
155 views

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
  • 159
16 votes
4 answers
780 views

$\sup \left\| A x + B y\right\|_2$ 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 ...
MWB's user avatar
  • 1,617
0 votes
1 answer
248 views

Quasiconvexity property of quasinorms

Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm. If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
Turbo's user avatar
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3 votes
0 answers
175 views

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
  • 546
0 votes
1 answer
83 views

On an error bound for matrix constraints

Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$. Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way ...
Turbo's user avatar
  • 13.7k
14 votes
5 answers
4k views

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 ...
Ivanov's user avatar
  • 157
1 vote
0 answers
75 views

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
  • 351
1 vote
1 answer
122 views

Applying backward shift operator on the composition operators on Hardy spaces

Let $H^2$ be the Hardy space. Let $K_\theta=H^2\ominus\theta H^2$, $\theta$ inner. Let $S$ be the shift operator on $H^2$. Its compression to $K_\theta$ is $S_\theta$. My question: do we have in the ...
Muath Karaki's user avatar
1 vote
1 answer
113 views

Reference request: Riesz operator over Hölder class

We have been told that the Riesz potential in $\mathbb{R}^d$, $I_{\alpha}(f)$, defined by $$I_{\alpha}(f)(x):= C\int_{\mathbb{R}^d} \frac{f(y)}{\left| x-y \right|^{d-\alpha}}\,\mathrm{d}y $$ has the ...
J. C. Cantero's user avatar
9 votes
1 answer
495 views

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 ...
Jochen Glueck's user avatar
3 votes
3 answers
914 views

Determinant of Jacobian and directional derivatives

I have a function $f: \Re^2 \to \Re^2$ and would like to understand why $$|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$$ that is, why the determinant of the Jacobian of $f$ at $...
D Blanc's user avatar
  • 39
-1 votes
1 answer
119 views

Hierarchies of Operator Norms [closed]

Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \...
Atransportconfusion's user avatar
2 votes
1 answer
107 views

Retractions for completely positive unital maps, and their effect on spectral diameter

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 $\Phi: \mathbf M_h(\mathbb C) \to \mathbf ...
Niel de Beaudrap's user avatar
4 votes
1 answer
197 views

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: \...
Niel de Beaudrap's user avatar
1 vote
0 answers
95 views

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
  • 163