# Questions tagged [operator-norms]

The operator-norms tag has no usage guidance.

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### Hilbert-Schmidt Operators

I am reading Brezis's book : Functional Analysis, Sobolev Spaces and Partial Differential Equations and trying to solve an exercise on Page 499. The question is as follows: Assume that $\Omega = (0, 1)...

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706 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 ...

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49 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:...

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38 views

### Dominating powers of a random matrix

Let $A_n$ be a (sequence of) random matrix such that $ A_n = (a_{ij})_{1 \leq i,j \leq n}$ and the $a_{ij}$ are iid, $\mathbb E\left[ a_{ij}\right] = 0 \quad E\left[ \vert a_{ij}\vert^2\right] = 1$.
...

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143 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 ...

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74 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 ...

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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 ...

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58 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 ...

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30 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 ...

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43 views

### Norm inequalities for operator valued matrices

I am looking for literature about operatorvalued matrices, especially norms and norm inequalites of these matrices (e.g. the entries are elements of the space of linear bounded operators $\mathcal{L}(...

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64 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 ...

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236 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 ...

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313 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 $...

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74 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 \...

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94 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 ...

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160 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: \...

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71 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 ...

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232 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 $\...

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103 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 ...

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299 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 $n \times n$ ...

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297 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+...

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128 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?
$$
\|A\| \le \|vec(A)\|_1
$$

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465 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 ...

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231 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 \...

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201 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 ...

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578 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}...

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74 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 $[A|b]$...

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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 $c$,...

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160 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 $b\in R$.
...

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393 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 ...

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282 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 ...

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224 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\|}{\|...

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125 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 $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...

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49 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 ...

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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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### 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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### 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 ...

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159 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 ...

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205 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 ...

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### $M_Λ(A) → A ⊗ M_Λ(C)$

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 ...

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178 views

### The definition of unitary fusion category

I just come across a definition of the unitary fusion category:
A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:
We have a Hilbert space structure on each ...

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235 views

### 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}\...

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358 views

### Elementary bound on operator norm on symmetric tensors: reference request

My education didn't really cover Tensors very well, so I'm getting stumped by a quite elementary question.
Let $T_k$ be a type k symmetric tensor. We can define the "operator norm" (or the induced ...

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298 views

### Comparison of Lp norm of matrix and its entry wise norm. [closed]

I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.

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654 views

### 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 $x_1,\ldots,x_m\in\mathcal{M}_k\,\,(m\in\mathbb{N}$ - ...

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596 views

### 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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622 views

### 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\...

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182 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 ...

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172 views

### Norm of a matrix operator with a special structure

Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that
$$\sum_{n=1}^{\infty}\alpha_n<\infty.$$
Question: Is there any chance to evaluate the operator norm of the matrix operator
$$C=...

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276 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)...