# Questions tagged [operator-norms]

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### 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 $n\times n$ ...
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### 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 ...
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### 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 ...
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### 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}) &=&...
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### 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?
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### 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 ...
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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### $M_Λ(A) → A ⊗ M_Λ(C)$
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 ...