# Tagged Questions

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### Operator norm and Hilbert Schmidt norm

I'm looking for a proof of $$||T||\leq ||T||_{HS},$$ for which it is sufficient to show ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
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### Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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### Immediate consequence of the definition of Operator Norm. Explain

||Av|| $\leq$ ||A||$_{op}$||v|| for every v $\in$ V I was wondering why this is true. Wikipedia says it's an immediate consequence of the definition but I just do not follow. I am using the ...
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### Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $A$ and $B$ ...
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### renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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### norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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### Integral Operator Theory on $L^2[0,1]$

Let K be the integral operator on l^2[0,1] defined by itex(t) = \int_0^t (t-s)f(s)\,ds[/itex] where 0\leq t\leq 1 Show that ||K|| <1 and that tex(t)= \int_0^t ...
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### Operators and Differential Equations

I have a question about m-dissipative operators. Thus let $T$ be an closed operator with dense domain. Then $(Tx,x)\leq0$ for all $x\in D(T)$ and $\lambda I-T$ surjective for all $\lambda>0$. I ...
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If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ... 1answer 225 views ### Bounded operator inverse, norm and spectrum I need help with an operator, I am not very good at functional analysis and need to find some properties of following operator:$X=C[(0,1)], A \in B(X); A[f(t)]=f(t^2)$1. I need to show that an ... 1answer 258 views ### Finding operator norm I have to solve the following problem: Find a norm of operator $$A:L^2[-\pi,\pi]\rightarrow L^2[-\pi,\pi]$$ given with $$Af(x)=\int_{-\pi}^{\pi} \cos^2{\left(\frac{x-t}{2}\right)}f(t) \,dt.$$ I ... 1answer 68 views ### product of bounded linear operators If I have 2 bounded linear operators$T_1,T_2$such that$T_2:X\rightarrow Y$and$T_1:Y\rightarrow Z$. I know that by boundedness,$||T_2(x)||\leq||T_2||\,||x||$and using the norm of$T$defined as ... 2answers 296 views ### What is the role of supremum in operator norm An operator norm is defined as$\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where$\|\cdot\|$is some norm on$\Bbb R^n$and$A\in M_n(\Bbb F)$, space of square matrices of dimension$n$over ... 2answers 68 views ### Linear functional$\mathscr{L}(E,F)$Let$\mathscr{L}(E,F)$denote the space of all linear functionals from$E \to F$. Let$\mathscr{C}(E,F)$denote the space of continuous linear functionals from$E \to F$. My question: How to prove ... 1answer 363 views ### Calculating the Norm of an operator in$L^2(0,1)$If I have the following operator for$H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate$||T||$, how do I go about doing this: I know that in$L^2(0,1)$we have that ... 1answer 110 views ### finding operator norm$T_N$How do find the operator norm of ,$T_N\colon c_0\to \Bbb R$given by$T_N(y):=\sum_{j=1}^Nx_jy_j$,when$N$is a integer . TIA 2answers 71 views ### Operator norm converging to 0 for certain condition Let$X$be a finite-dimensional normed space and$T_n : X \to X$a sequence of linear operators such that$\lim_nT_nx = 0$for all$x$in$X$. Prove that$\lim_n\|T_n\|=0$. 0answers 168 views ### The norm of an operator Let$\rho(x)$be a weight function in a unit sphere, such that \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ... 1answer 252 views ### norm equivalence Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ... 1answer 129 views ### Operator norm and spectrum Let$L$be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of$L$in terms of the spectrum of the positive operator$L^{\dagger}L$? ... 3answers 568 views ### How to find the norm of this bounded linear functional? Let$C^\prime[a,b]$denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval$[a,b]$in$\mathbf{R}$with the norm defined ... 2answers 262 views ### Norm of differentiation operator$Tf(t)=f^{'}$.. Consider$T:C^1[0, 1]\rightarrow C[0, 1]$given by$Tf=f'$where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and$\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove$\|T\|=1$? The inequality ... 1answer 86 views ### Norm of normal Operator A I just found the following equality$||A||=sup_{\lambda\in\sigma(A)} |\lambda|$My question: What exactly means$\sigma(A)$and why this is true ? I always thouht the only way to get the ... 1answer 223 views ### The Principle of Condensation of Singularities Let$X$,$Y$be Banach spaces and$\{T_{jk} : j,k \in\Bbb N\}$be bounded linear maps from$X$to$Y$. Suppose that for each$k$there exists$x\in X$such that$\sup\{\lVert T_{jk} x\rVert : j ...
How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$? 1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with (T_zx)(n)=z(n)\cdot ...