1
vote
1answer
59 views

Show that any compact set in $\mathbb{C}$ is the spectrum of an operator.

I have been looking around for an example of a general continuous (bounded) linear operator, who's spectrum is any compact set $K\subset\mathbb{C}$. I have seen an example, where we take the set ...
1
vote
1answer
27 views

What does the adjoint operator do? Is this Frechet derivative correct?

Problem statement Let $x \in l^2$ and $J(x) = \sum_{n = 1}^{+\infty} x_{2n - 1}^2$ Find first and second Frechet derivatives. Attempted solution Let's note that $J(x) = \sum_{n = ...
2
votes
1answer
52 views

show that the operator $T:l^2\rightarrow V$ is bounded

Let $V$ be the Banach space of all sequences $v=\{\eta_j\}_{j=1}^\infty$ such that $\lim_{j\rightarrow\infty}\eta_j$ exists. The norm on $V$ is given by $\|v\|=\sup_{j\in \mathbb{N}}\eta_j$. Consider ...
0
votes
1answer
42 views

Prove that — the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$

Prove that the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$.
0
votes
1answer
13 views

Image and Kernel of a certain bounded operator

Consider and Hilbert Space $X$, $T\in B(X)$ and a scalar $\mu$ s.t. $|\mu|=||T||$ By a simple argument I deduced that $\ker(\mu I- T)=\ker(\bar\mu I-T^*$) where $*$ denotes the adjoint. I am then ...
1
vote
0answers
43 views

Examples of operator theory on Hilbert space

$(1)$ If $T \in B(H)$ is self-adjoint and $T \neq 0$ then $T^n \neq 0$ $(a)$for $n=2,4,8,16,... (b)$ for every $n$ $(2)$ Show that any $T \in B(H)$ can be uniquely expressed as $T=T_1+iT_2$ ...
3
votes
1answer
66 views

Bounded Inverse Theorem

$A$ is a bounded linear operator from $X$ to $Y$ (both Banach spaces). Show that if there exists $k > 0$ such that $\|Ax\| \geq k\|x\|$, for all $x$ then $\operatorname{range}(A)\,$ is closed. My ...
0
votes
0answers
71 views

Proving that two operators are equal

So I'm trying to prove that there is an equivalence between $\langle \psi\mid T\varphi\rangle=\langle\psi \mid S\varphi\rangle$ and $\langle\varphi \mid T\varphi\rangle=\langle\varphi \mid ...
1
vote
1answer
38 views

Comparison of Symmetric Operators

The Problem: There is a unitary space $(V,<.,.>)$, $D \subseteq V $ a subspace and $ A,B : V \supseteq D\to V $ are two symmetric linear operators. Show that if: $<Ax , x> $$=$ $<Bx ...
1
vote
0answers
36 views

properties of integral operator $x^{-1}\int_0^xf(x,y)v(y)dy $

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) := ...
2
votes
2answers
137 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 ...
0
votes
3answers
68 views

Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
1
vote
1answer
48 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, ...
2
votes
1answer
145 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 $A \leq B$. ...
2
votes
1answer
50 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 $x$?
2
votes
1answer
30 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 ...
1
vote
1answer
55 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 ...
2
votes
2answers
80 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} ...
0
votes
0answers
28 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 ...
0
votes
0answers
38 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 ...
1
vote
0answers
48 views

Self-adjointness of differentiation operator

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 ...
2
votes
1answer
27 views

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 ...
2
votes
3answers
57 views

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: ...
9
votes
2answers
224 views

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 $x \in E$, ...
3
votes
2answers
135 views

Question about finding the norm of a bounded linear operator

Let H be a Hilbert space. Suppose $(i_k)_1^\infty$ is a complete orthonormal sequence in H. Let $a_k \in \mathbb{C}$ for $k \in \mathbb{N}$. Assume there is a bounded linear operator $T:H \rightarrow ...
0
votes
1answer
49 views

operator over inner product

Let $V$ be a finite-dimensional inner product space over $\mathbb{R}$ and let $u,v \in V$ be given. Define a linear operator $u\otimes v: V \rightarrow V$ by $(u\otimes v)x=<v,x>u$, where ...
2
votes
1answer
34 views

Question about putting an upper bound on a particular operator

So according to Wikipedia, given V(f)(t) = $\int_0^t f(s)ds$ where $f(s) \in L^2 (0,1)$ and $t \in (0,1)$. They say that $||V|| = \frac{2}{\pi}$ and I have seen the proof of this on a MSE post. The ...
5
votes
2answers
113 views

Question about a particular linear operator

Let A be a linear operator. $A: L^2(0,1) \rightarrow L^2(0,1)$ given by $Ag(a) = \int_0^a(a-x)g(x)dx$ where $a \in (0,1)$. This is the integral operator, and we know ||A|| < 1 which is easy to ...
2
votes
2answers
66 views

Spectrum of an element

I'm having a little trouble calculating the spectrum of an element: specifically, the element $f(x) = \frac{1}{x}$, as an element of the bounded continuous functions from $[1, \infty)$ with pointwise ...
1
vote
1answer
95 views

Lax-Milgram problem

I am trying to solve this problem: Let $H$ a Hilbert space, $A:H\times H\rightarrow\mathbb{R}$ a bilinear form, bounded and $H$-elliptic, and $F\in H^{\prime}$ ($H^{\prime}$ = dual space). Besides, ...
1
vote
0answers
118 views

Compact integral operator

I have this exercise and I don't know how to solve the last question. In the following $a,b$ are two real numbers such that $a<b$ ,$E=C([a,b],\mathbb{R})$ with the norm $||.||_0$ given by ...
0
votes
1answer
131 views

Invariant space of linear transformation

Let $V$ be a vector space of a finite nonzero dimension $n$ over some field. Let $T$ be a linear transformation of $V$, such that $T$ is nonzero and not one-to one. (a)Give a $T$-invariant linear ...
2
votes
1answer
118 views

Prove that the limit exist II

First question was here. I add one new condition. 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 ...
4
votes
1answer
76 views

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 ...
1
vote
1answer
161 views

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 ...
4
votes
1answer
54 views

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 ...
1
vote
1answer
81 views

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{% ...
3
votes
1answer
56 views

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: ...
1
vote
0answers
30 views

To prove that an operator is bounded [duplicate]

I have this problem: Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space on $\mathbb{C}$ and $A:H\rightarrow H$ a linear operator such that $$\langle A(x), y\rangle\ =\ \langle x, ...
1
vote
0answers
67 views

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) ...
1
vote
0answers
117 views

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 $Au=v,$ ...
2
votes
1answer
113 views

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 ...
1
vote
1answer
63 views

Orthogonality & Adjoint Operator

I am trying to prove this simple statement left to the reader in Br├ęzis's book. Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
5
votes
1answer
146 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
4
votes
0answers
143 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
5
votes
2answers
348 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
1
vote
1answer
138 views

hoffman-and-kunze [chapter 8.2][question 4]

The question: Let Y be a finite-dimensional inner product space and T a linear operator on Y. Show that the range of T* is the orthogonal complement of the null space of T Think i got one way: took ...
5
votes
1answer
292 views

Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
4
votes
0answers
144 views

Prove that the integral operator is bounded [duplicate]

Prove that the following operator is bounded on $L^{2}(0, \infty)$: $Af(x)$ = $\frac{1}{\pi} \int_{0}^{\infty} \frac{f(y)}{x+y}dy$ with $||A|| \le 1$. Attempt at Solution It can be shown that: ...
0
votes
1answer
60 views

problem related to tensor product on Hilbert spaces

Let $K$ and $H$ be Hilbert spaces. Let $\{e_i:i\in I\}$ be an orthogonal basis of $H$. Define $$ U_i:K\to K\overset{.}{\otimes} H: x\mapsto x\overset{.}{\otimes} e_i $$ Assume ...