The operator-norms tag has no wiki summary.

**0**

votes

**0**answers

60 views

### Bounding multiplications of PSD random matrices

Consider the following setup,
$(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$.
The ...

**0**

votes

**1**answer

40 views

### Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows
\begin{eqnarray*}
||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...

**0**

votes

**0**answers

47 views

### Relation between the block maximum norm and the Euclidean norm of a matrix

I am trying to give answer to the following question:
Let's define de block maximum norm of a $N*M \times N*M$ matrix as
\begin{eqnarray*}
\parallel A \parallel_b = max_{x \neq 0} [ \parallel A x ...

**0**

votes

**0**answers

24 views

### Proximal operator of modified L1 matrix norm

In literature proximal operator $prox_{\lambda f} : R^n \rightarrow R^n$ of $f$ is defined as:
$prox_{\lambda f}(V) = argmin(X) (f(X) + (1/2 \lambda)||X-V||^2_2)$
Consider now $g(X) = ...

**3**

votes

**1**answer

140 views

### 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 ...

**2**

votes

**1**answer

81 views

### Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...

**8**

votes

**2**answers

310 views

### 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 ...

**6**

votes

**2**answers

306 views

### $\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

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 ...

**10**

votes

**1**answer

321 views

### 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
$$
...

**10**

votes

**1**answer

304 views

### 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 ...

**15**

votes

**1**answer

470 views

### A matrix completion problem

In their paper, "Corners of Normal Matrices," R. Bhatia and M.D. Choi ask the following question: Given a matrix pair $(B,C)$ where $B,C∈M_n$, does there exist matrices $A,D ∈ M_n$ such that the block ...

**5**

votes

**1**answer

149 views

### 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 ...

**1**

vote

**0**answers

88 views

### All Unitarily Invariant Minkowski Norms on $\mathfrak{su}(N)$

It is well known that the some matrix norms satisfying:
$\|A\| = \|UAV\|$, $\forall A \in \mathbb{C}^{N \times N}, \forall U,V \in U(N)$
are the Schatten-p norms:
$\|A\|_p= \left( \sum_{k=0}^{N-1} ...

**2**

votes

**0**answers

50 views

### Rank of the smallest rank-reducing perturbation of a general matrix

Let $A$ be a general $m \times n$ matrix (not necessarily square, and not necessarily of full rank). Let $\| \cdot \|$ be a norm for $m \times n$ matrices that is induced from a norm for $m$ vectors ...

**0**

votes

**2**answers

143 views

### Matrix-Norm aquivalence with p-Norm [closed]

Let $A$ be a square Matrix and $||\cdot ||_p$ the induced Matrixnorm for $1 \leq p \leq \infty$. Is it true that
$$||A||_p\leq \max(||A||_1,||A||_{\infty})?$$
For $p=2$ the answer is yes because ...

**0**

votes

**1**answer

269 views

### solving trace norm equality [closed]

Problem Formulation
under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$.
...

**6**

votes

**1**answer

189 views

### 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} ...

**1**

vote

**0**answers

87 views

### expansion with respect to p-norms for p other than 2

Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$.
Let $x \in ...

**1**

vote

**0**answers

179 views

### Tensors with low spectral norm

Consider a tensor $T$ with six indices, $T_{(ii')(jj')(kk')}$, where each index goes from $1$ to $n$. We can think of $T$ as a linear map from $\mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$ ...

**4**

votes

**1**answer

434 views

### 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 ...

**4**

votes

**2**answers

1k views

### bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?

Let $A_i$ with $i=1,\dots,N$ 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 $\sum_{i=1}^N a_i = 1$.
Claim:
...

**3**

votes

**1**answer

335 views

### injective tensor norms for real tensors

If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be
$$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1}
...

**3**

votes

**2**answers

701 views

### 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.

**2**

votes

**2**answers

231 views

### Smooth dependence of the spectrum on the operator

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter.
To clarify, suppose I have a 1-parameter family $T_h$ of ...

**5**

votes

**1**answer

431 views

### 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 ...

**2**

votes

**1**answer

314 views

### Equivalence constante between factorization norm and trace norm

The factorization norm, sometimes also called $\gamma_2$ norm play an important role in (quantum) communication complexity and is defined for a $n\times n$ matrix $A$ by:
$\gamma_2(A) = \max || A ...

**3**

votes

**2**answers

846 views

### 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 ...

**3**

votes

**0**answers

161 views

### 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
...

**5**

votes

**1**answer

406 views

### 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 ...

**4**

votes

**3**answers

1k views

### 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} ...

**7**

votes

**1**answer

862 views

### 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 ...

**4**

votes

**0**answers

312 views

### 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 ...

**2**

votes

**0**answers

265 views

### Do Isometry Groups Tell Us How Difficult Norms are to Compute?

The question: Consider two norms N1 and N2 on the space of n-by-n complex matrices. N1 and N2 have the same isometry group and computing N1 is NP-HARD. Does it follow that computing N2 is NP-HARD as ...

**17**

votes

**1**answer

530 views

### 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 ...

**3**

votes

**1**answer

261 views

### Bounds on operator 2-norms on partial traces of linearly related operators

Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on ...