Questions tagged [operator-norms]
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123 questions
0 votes
0 answers
106 views
When is −∥x−y∥+∥A(x−y)∥ a CPD kernel ? - Difference of CPD radial kernels remains CPD (Bernstein functions)
I’m studying a family of translation-invariant kernels built from two norms and would like to understand exactly when their difference remains conditionally positive-definite. In particular, we focus ...
- 1
0 votes
0 answers
125 views
Matrix norm inequality
I have two $n\times n$ matrices A and B where $A$ is positive-definite. Is there any matrix norm such that if $||B|| \le c$, I could conclude $B^T A B \preceq c^2A$ in the Loewner order?
- 1
2 votes
0 answers
102 views
Does b.a.u. convergence imply a.u. convergence for $L_p$-norm convergent sequences, $p \geq 2$?
A wee recap of the definition(s): Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a normal semifinite faithful trace $\tau$. We denote by $L_p(\mathcal{M})$ the noncommutative $L_p$...
- 133
0 votes
0 answers
109 views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported. I am wondering if we can ...
- 301
3 votes
0 answers
133 views
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function $$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$ where $\lambda_{\max}$ denotes the largest eigenvalue....
- 123
0 votes
1 answer
219 views
Counterexample wanted: Banach space but not BK-space
What is an example of a Banach space that is not a BK-space? A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = ...
- 1
12 votes
0 answers
840 views
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 ...
- 4,493
0 votes
1 answer
565 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 $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
- 742
13 votes
0 answers
204 views
Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$. If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
- 536
3 votes
2 answers
2k views
Is the matrix induced L1-norm greater than the induced L2-norm?
For $A \in \mathbb R^{m \times n}$ and the induced norms: $$ \| 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 = \...
- 165
1 vote
0 answers
310 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 ...
- 97
1 vote
0 answers
326 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. ...
- 11
0 votes
0 answers
128 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....
- 151
3 votes
0 answers
415 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 ...
- 223
0 votes
1 answer
362 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: \...
- 131
1 vote
0 answers
184 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 ...
- 131
5 votes
1 answer
747 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. $$ \left\lVert\pmatrix{a&...
- 20.1k
1 vote
0 answers
105 views
How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]
Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$. I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$. My belief is that this is true is motivated by empirical ...
- 11
3 votes
1 answer
648 views
Trace norm of operators obtained by restricting the matrix of a trace class operator
Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...
- 243
1 vote
0 answers
72 views
Row-wise conjugation of completely bounded map by group action
Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...
- 3,128
9 votes
1 answer
376 views
Continuously varying norms
Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number $$ d =...
- 1,404
4 votes
1 answer
281 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 ...
1 vote
1 answer
207 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\...
- 1,980
2 votes
1 answer
849 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 ...
- 607
1 vote
1 answer
219 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-...
- 343
1 vote
1 answer
166 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 $u(\lambda x)=|\lambda|u(x)$ ...
- 4,132
7 votes
1 answer
3k 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 ...
- 71
9 votes
1 answer
626 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 ...
- 111
6 votes
1 answer
316 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, ...
- 201
3 votes
0 answers
175 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 $n\times n$ ...
- 81
3 votes
1 answer
141 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 ...
- 53.1k
2 votes
0 answers
173 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 ...
- 121
0 votes
1 answer
98 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}) &=&...
- 1,154
1 vote
0 answers
383 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?
- 51
7 votes
2 answers
2k 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 $$T_1\otimes ...
- 71
5 votes
0 answers
148 views
Schur norm of weighted Cauchy matrix
The Schur norm of a matrix $A$ is defined to be $\|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$, where $\|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value. Let $a_1,\ldots, ...
- 81
0 votes
1 answer
196 views
Does $\{\left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right||\varphi\in\{\psi\}^{\perp}\}$ split $\mathfrak{S}_1$?
Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}...
user141436
2 votes
2 answers
482 views
Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different
In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
- 741
1 vote
1 answer
1k views
Relationship between $2 \to 2$ norm and $\infty \to 2$ norm [closed]
I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is. E.g., the trivial result is that for matrix $A$ with ...
- 181
1 vote
1 answer
790 views
Minimal value of matrix norm induced by a norm
Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by $$ \| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}} $$ where the matrix $A$ is interpreted as an ...
- 252
2 votes
1 answer
2k views
Inequality for the operator norm of a product of matrices
I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that $$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
- 459
5 votes
0 answers
201 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 ...
- 159
16 votes
4 answers
861 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 ...
- 1,727
0 votes
1 answer
342 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:/...
- 1
3 votes
0 answers
196 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 ...
- 576
0 votes
1 answer
118 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 ...
- 1
15 votes
5 answers
5k 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 ...
- 167
1 vote
0 answers
89 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 ...
- 385
1 vote
1 answer
151 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 ...
- 103
1 vote
1 answer
223 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 ...