Questions tagged [operator-norms]

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2
votes
0answers
66 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 $n\times n$ ...
2
votes
1answer
96 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 ...
2
votes
0answers
148 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 ...
0
votes
1answer
56 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
vote
0answers
56 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?
4
votes
1answer
289 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 ...
5
votes
0answers
66 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, ...
0
votes
1answer
155 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}...
3
votes
1answer
245 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 $\...
0
votes
1answer
96 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 ...
1
vote
1answer
148 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 ...
1
vote
1answer
257 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,...
4
votes
0answers
109 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 ...
16
votes
4answers
733 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 ...
0
votes
1answer
90 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:...
3
votes
0answers
156 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 ...
0
votes
1answer
77 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 ...
14
votes
5answers
2k 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 ...
1
vote
0answers
71 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 ...
1
vote
1answer
86 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 ...
1
vote
1answer
74 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 ...
9
votes
1answer
316 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 ...
3
votes
2answers
442 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 $...
-1
votes
1answer
75 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 \...
2
votes
1answer
99 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 ...
4
votes
1answer
165 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: \...
1
vote
0answers
76 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 ...
6
votes
1answer
264 views

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 $\...
2
votes
0answers
152 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 ...
6
votes
1answer
564 views

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 $n \times n$ ...
6
votes
3answers
330 views

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+...
-4
votes
1answer
171 views

Matrix norms: operator norm vs vectorized $L_1$-norm? [closed]

Let for a square matrix $A$ the operator norm $\|A\|$, and vectorized $L_1$-norm $$ \|vec(A)\|_1 = \sum_{i,j} |A_{ij}| $$ Do we have the following? $$ \|A\| \le \|vec(A)\|_1 $$
3
votes
1answer
613 views

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 ...
3
votes
0answers
273 views

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 \...
0
votes
1answer
283 views

Bounds on spectral norm of difference of powers of linear operators

Given bounded linear operators $A$ and $B$, with $||A-B||<\epsilon$, what is the tightest upper bound on $||A^n-B^n||$, for positive integer $n$? Additionally, for my specific situation, $A$ is ...
12
votes
2answers
822 views

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}...
2
votes
0answers
78 views

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 $[A|b]$...
4
votes
0answers
78 views

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 $c$,...
1
vote
1answer
164 views

Spectrum on an unbounded operator

Consider the operator $T_{c}=-\frac{d}{dx^{2}}+ c x^{2}$ with $c\in C^{*}$, $Re(c)>0$ defined on its domain $D_{c}=\{u\in L^{2}; T_{c}(u)\in L^{2}\}$. Put $c=a+ib$ avec $a>0$ et $b\in R$. ...
-1
votes
1answer
496 views

A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result: For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...
7
votes
0answers
360 views

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 ...
8
votes
2answers
276 views

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
vote
1answer
142 views

Matrix norm inequality for C*-Algebras [closed]

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...
0
votes
1answer
51 views

Simplify the expression of $ T^+$ for an unbounded operator $T$?

For a negative unbounded operator $T$, what equals the operator $$ T^+ = \left[\frac{1}{2}(|T| + T) \right]^{**},$$ where $|T|= (T^2)^{1/2}$ and $A^{**} $ is the minimal closed extension of an ...
5
votes
2answers
1k views

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 $(...
29
votes
0answers
1k views

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 ...
3
votes
2answers
2k views

Weak convergence implies norm convergence for trace class operators?

It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. here. Because of the typical analogies of the Schatten ideals $C_p \subset B(H)$ (where $H$ is a Hilbert ...
7
votes
2answers
167 views

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 ...
-1
votes
1answer
214 views

How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]

(Sorry, I do hate editing this many many times but let me try the last time) Gelfand's formula says that $$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$ I am wondering whether there is any ...
3
votes
1answer
132 views

$M_Λ(A) → A ⊗ M_Λ(C)$

I saw this in here. Let $A$ be a Banach algebra, and let $\Lambda$ be a non-empty set. We denote by $M_\Lambda(A)$ be the set of $\Lambda\times\Lambda$ matrices $(a_{ij})_{i,j\in\Lambda}$ with entries ...