Tagged Questions

1
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0answers
27 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
26 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
33 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $0 \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ where ...
2
votes
1answer
48 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
53 views

Orthogonality & Adjoint Operator

I am trying to prove this simple statement left to the reader in Brézis's book. Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
2
votes
1answer
69 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 ...
2
votes
0answers
113 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 ...
2
votes
1answer
97 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
56 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
175 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
120 views

Prove that the integral operator is bounded

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
54 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 ...
2
votes
1answer
45 views

Spectrum in Hilbert space

Let $H$ be a Hilbert space and $T\in B(H)$ be normal. Want to show that if $\lambda$ is an isoloated point in $\sigma(T)$ then $\lambda$ is an eigenvalue of $T$.I have no idea how to do this one. ...
4
votes
2answers
133 views

Functional analysis-Closed graph therem

Let $ X$, $ Y$, $ Z$, be Banach spaces and let $ T:X\to Y $ and $ S:Y\to Z $ be linear transformations.Suppose that $S$ is Bounded and injective and that $ S \circ T $ is bounded.Prove that $T $ is ...
-1
votes
1answer
106 views

Functional analysis - bounded linear transformation

Let $ \mathcal{H} $ be a Hilbert space, and let $ T: \mathcal{H} \to \mathcal{H} $ be such that $ \langle x,Ty \rangle = \langle Tx,y \rangle $ for all $ x,y \in \mathcal{H} $. How can one show that ...
3
votes
1answer
108 views

cyclic vector exists for symmetric operator iff there no repeated eigenvalues

Considering a symmetric operator $A$ acting on a finite dimensional Hilbert space $H$, we say $x\in H$ is a cyclic vector for $A$ if the set of finite linear combinations of $\{A^n x:n=0,1,2,...\}$ is ...
2
votes
1answer
59 views

Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
3
votes
1answer
71 views

Does $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for bounded operators on Hilbert space?

If $A$ is a bounded linear operator on a Hilbert space $H$ is it true that $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for all $x\in H$? If not, can we at least establish inequality in one ...
2
votes
0answers
95 views

(SOLVED) Adjoint of Frechet derivative (involving gradient operator)

I need some help with a problem (a homework/programming exercise) regarding the adjoint operator of the Frechet derivative of an operator. I have the forward operator $ F(a) = L_a ^{-1}f $ where ...
3
votes
2answers
116 views

Proof that certain operators are compact

I want to examine which of the following operators $T \colon C[0,1] \to C[0,1]$. are compact, by some I think I got the argument, but others I have no idea. a) $Tx(t) = x(t^2)$ Guess it is ...
1
vote
2answers
139 views

operator norm of this multiplier operator

I am having some trouble with some basic properties of a given operator. Firstly, the operator T is defined as taking the fourier inverse transform of the function ...
2
votes
1answer
91 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
1
vote
1answer
205 views

In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of $n$ orthonormal basis elements converge?

Could anyone help me with this problem? I don't know where to start. Let $\{ e_n \}_{n=1}^\infty$ be an orthonormal basis in a separable Hilbert space $H$. Denote by $P_n$ the orthogonal ...
0
votes
1answer
31 views

Conditions for additivity of the trace of projections

I came up with the following problem: Let $\Pi_A$ and $\Pi_B$ be two projection operators on two disjoint subspaces of a certain Hilbert space $\mathcal H$ and let $\rho$ be unit trace, positive, ...
3
votes
1answer
460 views

Compactness and boundedness of integral operator

I got some trouble with my homework question : Let $B$ be the unit ball in $\mathbb{R}^d$, and let $T$ be an integral operatpor on $L^2(B)$ with kernel $K(x,y)$. Suppose that $\sup_x \int_B ...
3
votes
1answer
98 views

Operator norm of the sum of a finite collection of bounded linear operator

I recently got some difficulty with my homework question. The question is: Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le ...
2
votes
1answer
206 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
2
votes
1answer
222 views

A compact operator is completely continuous.

I have a question. If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous. A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
3
votes
1answer
51 views

Convergent operatorial series

An exercise I was doing asks (among other things) for the values of $z\in\mathbb{C}$ for which the following (operatorial) series converges absolutely: $$\sum_{n=0}^{\infty}z^nA^n$$ where $A$ is an ...
1
vote
1answer
40 views

Well definededness of integration with respect to a projection valued measure

Let $(X,\mathcal{F})$ be a measurable space and let $E:\mathcal{F}\to\mathscr{B(H)}$ be a spectral measure. Let $\phi\in B(X)$ be a simple function whose image is ...
0
votes
1answer
77 views

Condition for frame of $L_2$

Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions $$ f_{m,k}=f^m(x-2^k) $$ Under which conditions $\{f_{m,k}\}$ would be a ...
6
votes
2answers
216 views

Proving $A: l_2 \to l_2$ is a bounded operator

Let us consider the following linear operator acting on $l_2$: $$ A(x_1,x_2,x_3,\ldots) ~\colon=~ \left(x_1,\frac{x_1+x_2}{2},\frac{x_1+x_2+x_3}{3},\ldots\right) $$ I need to show that $A$ is a ...
6
votes
2answers
278 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
3
votes
1answer
220 views

Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
3
votes
1answer
234 views

Spectrum of a “quasi” right shift operator

Let $\mathcal{H}$ be a Hilbert space and let {$e_j$}$_{j\in \mathbb{Z}}$ be an orthonormal basis for $\mathcal{H}$. Define a linear operator $T$ on $\mathcal{H}$ by $T(e_0) = 0$ and $T(e_j) = e_{j+1}$ ...
0
votes
1answer
175 views

When is this Self-adjoint?

When is the following operator self-adjoint?(Is there a difference between self-adjoint and Hermitian?) $O:= \sum_{n=0}^4 f_n(x){d^n\over dx^n}$ subjected to boundary conditions ...
6
votes
2answers
393 views

Finding all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy 2 conditions

As above, I'm trying to find all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy the following 2 conditions: I) $Lf \, \geq \, 0$ for all non-negative $f\in C([0,1])$. II) $Lf = f$ for ...