# Questions tagged [operator-norms]

The operator-norms tag has no usage guidance.

110
questions

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### Application of the Frechet derivative [closed]

$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that:
\...

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0
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### Equivalence constants for induced matrix norms

Disclaimer: I asked this question beforehand on mathematics stack exchange, but I think it is better suited for this site
Given two sets $P_i\in\mathbb{R}^s$, bounded, convex, with non-empty interior ...

5
votes

1
answer

166
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### Euclidean norms of matrices

Let us consider the euclidean norm on $\mathbf{R}^2$. After some computations, I have obtained the following expression for the associated operator norm on 2 by 2 matrices.
$$
\left\lVert\pmatrix{a&...

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0
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130
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### Proof $(AB)^{\top}=B^{\top}A^{\top}$ via $(AB)^{-1}=B^{-1}A^{-1}$ and density argument. Unification of $(AB)^{\star}=B^{\star}A^{\star}$ [closed]

Question: $\mathrm{cl}\, \mathrm{O}_n(\mathbb{R})=\mathrm{GL}_n(\mathbb{R})$?
Context: Let $\mathrm{GL}_n(\mathbb{R})$ be a set of nonzero determinant real matrices dimension $n\times n$, and $\mathrm{...

1
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### How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]

Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$.
I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$.
My belief is that this is true is motivated by empirical ...

3
votes

1
answer

276
views

### Trace norm of operators obtained by restricting the matrix of a trace class operator

Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...

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### Row-wise conjugation of completely bounded map by group action

Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...

9
votes

1
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252
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### Continuously varying norms

Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d =...

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votes

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138
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### Operator norm of a masked SDP matrix

Let $\Sigma$ be a $d\times d$ semi-definite positive matrix (SDP). Let $I\subset\{1,\ldots, d\}\times \{1, \ldots, d\}$ be a symmetric subset of indices (i.e. if $(p,q)\in I$ then $(q,p)\in I$). We ...

1
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1
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157
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### Some estimates on tensor norms

Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...

1
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1
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182
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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 [1] authors define the following ...

1
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1
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175
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### 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-...

1
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1
answer

110
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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 $u(\lambda x)=|\lambda|u(x)$ ...

4
votes

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625
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### 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 ...

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1
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### 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 ...

6
votes

1
answer

257
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### 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, ...

2
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0
answers

107
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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 $n\times n$ ...

3
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1
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108
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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 ...

2
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150
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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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169
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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?

6
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780
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### 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 ...

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### 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, ...

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1
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166
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### 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}...

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### 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 $\...

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### 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
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1
answer

282
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### 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

1
answer

873
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### 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,...

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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 ...

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votes

4
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### $\sup \left\| A x + B y\right\|_2$ 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
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1
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139
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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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0
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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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1
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80
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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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votes

5
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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 ...

1
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0
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73
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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 ...

1
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1
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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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1
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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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1
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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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2
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662
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### 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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votes

1
answer

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### 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
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1
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### 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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### 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
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0
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### 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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1
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319
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### 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

0
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220
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### 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
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1
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890
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### 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

3
answers

391
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### 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+...

3
votes

1
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840
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### 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

0
answers

286
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### 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

1
answer

399
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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 ...