# Tagged Questions

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here we have two cases to study $(1)$ let us fix any $f \in C^{1}[ [0,1] \times [0,1]]$ ($k \neq 0$). Set $$[T(v)](x) := x^{-1}\int_0^xf(x,y)v(y)dy$$ for any $x \neq 0$ otherwise $[T(v)](0) := ... 2answers 277 views ### Norm of a matrix equals greatest eigenvalue How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let$A$be a symmetric$n \times n$matrix. Consider$A$as an ... 3answers 69 views ### Compute the norm of matrix Let$M$be$n\times n$matrix, consisting entirely of 1's. Show, that$\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$. 1answer 49 views ### Relation of norms of matrices Let$A$be$m \times n$matrix. Let$B=\frac 1n AA^*$, where$A^*$is a transposed matrix. Let$X_i, I\leq m$be row-vectors of$A$. Show $$\|B\|=\frac 1n \|A\|^2\geq \max_{i\leq m}|X_i|,$$ Where, ... 1answer 177 views ### Inverse of positive operators Does anyone know how to show this? Let$H$be a Hilbert space and$A$,$B$bounded positive operators defined on$H$such that$A^{-1}: H \rightarrow H$exists and hence bounded and$A \leq B$. ... 1answer 51 views ### Existence of bounded linear operator with kernel reduced to$\{0\}$If$X$and$Y$are normed spaces, why there must exist a bounded linear operator$T$from$X$to$Y$such that$T(x)$is not equal to$0$for all non-zero$x$? 1answer 31 views ### About convergence of$(T_nR_n)$when$(T_n),(R_n) \subset B(X)$Let$X$be a Banach space and$(T_n),(R_n) \subset B(X)$. (a) Prove that if$(T_n)$converges strongly and$(R_n)$converges strongly or uniformly, then$(T_nR_n)$converges strongly (b) Prove that ... 1answer 59 views ### Identity plus finite rank has index$0$I'm supposed to prove the strong Fredholm alternative in the form $$\text{Ind}(1-K)=0$$ for any compact operator$K:H\to H$where$H$is a Hilbert space and $$\text{Ind}(T):=\text{dim Ker }T+\text{dim ... 2answers 87 views ### Show that the operator is bounded in L_p Consider the operator C, acting on functions f on the unit circle S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\} by the rule$$ (Cf)(z) = \frac{1}{2\pi i} ... 0answers 29 views ### About weak-measurability in L(X,Y) I'm starting a project about vector- (Banach) valued funtions and measures, and I know some of the basic definitions (measurable,weakly measurable...). The functions of study are ... 0answers 40 views ### tight frame for$\mathbb{C}^N$I have a question to ask Prove that if$K\in\mathbb{Z}-\{0\}$, then$\{\phi_p[n]=\exp(i2\pi pn/(KN))\}_{0\leq p<KN}$is a tight frame of$\mathbb{C}^N$, i.e.$\sum_{k}|\langle f,\phi_p\rangle ...
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Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
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### Condition on spectrum of T

Let $T$ $\in \mathfrak{B}(\mathbb{H})$ be normal. Let $A$ be the closed subalgebra generated by $T$, $T^{*}$ and $I$. Suppose $T$ can be approximated in norm by finite linear combinations of ...
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### Operator attains norm

We have the following linear and well-defined mapping $T_{a}(b):=\sum_{j=1}^{\infty}a_{j}b_{j}$, with $\{a_{j}\}\in \ell^{1}$ and $\{b_{j}\} \in c_{0}$. Show that $\|T_{a}\|=\|a\|_{1}$. My work: ...
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### Why is such an operator continuous?

These two questions were in one question of a list of exercises. Let $E$ be a Banach space and $T : E \longrightarrow E^*$ be linear. If $\langle T(x),x \rangle \geq 0$ holds for all $x \in E$, ...
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### Prove that the limit exist

Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) operator, $H$ is a Hilbert space (not necessary). Suppose that $\forall h \in H \Rightarrow Th=h$ $T_n$ - a sequence of ...
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### Operator norm, convolution and Gauss-Weierstrass kernel

Let be $g_{t}(x)=\frac{1}{\left( 4\pi t\right) ^{\frac{1}{2}}}e^{-\frac{x^{2}% }{4t}},t>0,x\epsilon %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ Gauss-Weierstrass kernel. For ...
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### An operator between $\mathcal{L}(X, Y)$ and $\mathcal{L}(Y, X)$

Please, I need help with this problem. Let $X$, $Y$ be two vector normed spaces. Let $A_0\in\mathcal{L}(X, Y)$ such that $A^{-1}_0\in\mathcal{L}(Y,X)$. Show that there's an operator ...
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### Norm operator and compactness

For the operator $U\colon \ell_{p}\to\ell_{p},\;\left( 1\leqslant p<\infty \right) :$ \begin{equation*} Ux=U\left( x_{1},x_{2},\dots \right) =\left( 0,x_{1},\frac{x_{2}}{2},\frac{% ...
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### A simple adjoint operator question

I'm trying to solve this problem: Let $\Omega$ a bounded open of $\mathbb{R}$, consider the Hilbert real spaces $X = L^2(\Omega)$ and $Y = \mathbb{R}^{2\times 2}$, with the inner products: ...
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### Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
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### Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
### Extension of a Bounded Operator on $L^p$ to $L^r$
Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...