# Questions tagged [operator-norms]

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### 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  authors define the following ...
172 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-...
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### 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 $u(\lambda x)=|\lambda|u(x)$ ...
262 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 ...
310 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 ...
202 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, ...
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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 $n\times n$ ...
108 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 ...
149 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 ...
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}) &=&...
93 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?
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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 ...
749 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 ...
105 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:...
166 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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### 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 ...
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 ...
73 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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### 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 ...
348 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 ...
538 views

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 $... 1answer 77 views ### Hierarchies of Operator Norms [closed] Given some linear operator$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 \... 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 ... 1answer 170 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: \... 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 ... 1answer 286 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 \... 0answers 174 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 ... 1answer 740 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 ... 3answers 346 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+... 1answer 187 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? $$\|A\| \le \|vec(A)\|_1$$ 1answer 732 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 ... 0answers 280 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 \...
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 ...