Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

learn more… | top users | synonyms

2
votes
1answer
31 views

Finding eigenfunctions and eigenvalues to Sturm-Liouville operator

I'm struggling to understand how to find the associated eigenfunctions and eigenvalues of a differential operator in Sturm-Liouville form. For instance, one question that I am trying to solve is the ...
1
vote
1answer
29 views

Partial Isometries: Characterizations

Any partial isometry satisfies: $$\Omega\Omega^*\Omega=\Omega$$ From this, one derives projections: $$\Omega^*\Omega,\Omega\Omega^*$$ Conversely, given projections: $$\Omega^*\Omega,\Omega\Omega^*$$ ...
1
vote
1answer
60 views

Example of a wot convergent net but not $\sigma -$ weak convergent

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ ...
2
votes
1answer
36 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
2
votes
1answer
70 views

Show self-adjointness on manifold

I was wondering if anybody here knows how to show that $ - \Delta $ is self-adjoint on $H^2( \mathbb{S}^2)$? I read that it is a rather cumbersome calculation, but I also don't see how one should ...
1
vote
1answer
37 views

Spectrum of unitary transform

Let $T: \operatorname{dom}(T) \rightarrow H$ be self-adjoint, then $U(T):=(T+i)(T-i)^{-1}$ is defined and unitary( this is clear to me). Furthermore, we have that $\sigma(U(T)):= \overline{\{ t; ...
1
vote
0answers
31 views

Diagonalisability of Self-Adjoint Operators for Non-Symmetric Metrics

Let $V$ be a finite dimensional vector space and $(\cdot,\cdot)$ a non-degenerate bilinear form. When $(\cdot,\cdot)$ is symmetric, every self-adjoint operator on $V$ is diagonalisable. What happens ...
1
vote
1answer
19 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
0
votes
1answer
44 views

Møller Operators: Unitary Equivalence

Reference This is taken out of M. Reed and B. Simon, Scattering Theory. Problem Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the ...
-1
votes
1answer
27 views

Møller Operators: Absolutely Continuous Subspaces [closed]

Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the Møller operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau ...
3
votes
1answer
40 views

Comparing weak and weak operator topology

We can compare topologies on $B(H)$. For instance, Sot topology is stronger than wot topology or $\sigma-$ weak topology is equivalent to weak* topology. I would like to compare wot topology and weak ...
4
votes
1answer
55 views

A question about local convexity of the weak operator topology

By definition, I know a locally convex space is a topological vector space whose topology is defined by a family of seminorms $\cal P$ such that $$\bigcap_{p\in{\cal P}}\{x\colon p(x)=0\}=\{0\}.$$ ...
1
vote
1answer
18 views

Nonnormal Operator: Empty Spectrum

Are there operators on Hilbert space having empty spectrum? (Surely, for Banach spaces they do exists.) Necessarily, they must be closed and can't be normal.
2
votes
1answer
32 views

How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
2
votes
2answers
31 views

Show existence and uniqueness of integral equality with neumann-series

I want to show that for $$x(s)-\int_0^12rs\cdot x(r)dr=\sin(\pi s)$$ there exists exactly one solution $x \in C^0([0,1],\mathbb R)$.
1
vote
1answer
29 views

Positive operator has a positive spectrum?

Let $T : \operatorname{dom}(T) \rightarrow H $ be a positive self-adjoint operator, is it then true that $\sigma(T) \subset [0,\infty)$? This is something that sounds natural and I guess that it is ...
0
votes
2answers
39 views

Spectral Measures: Spectrum vs. Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
0
votes
0answers
21 views

Integro-differential operator

Good morning everybody, recently I came across the following question: is it possible to characterize (i.e. giving differential conditions which are necessary AND sufficient) the solutions of the ...
2
votes
0answers
82 views

Translational invariance and zero eigenvalue

Page 2 (506), line 18 of http://www-personal.umich.edu/~orosz/articles/NonlinScipublished.pdf says that "The presence of translational symmetry in the nonlinear equations gives rise to a relevant ...
0
votes
1answer
50 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
-2
votes
1answer
31 views

bounded linear operators between Banach spaces [closed]

Definition 1 Let $E$ and $F$ be Banach spaces. An operator $T$ is called bounded linear from $E$ into $F$ if there exist a positive constant $C$ such that $$\left\|T x\right\|\leq C ...
2
votes
1answer
71 views

Limit point / limit circle and self-adjointness

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
1
vote
1answer
28 views

Confusion about the definition of self adjoint and formally self-adjoint

I have some confusion about the definition of self-adjoint operators and formally self-adjoint operators. Let me write down the background information. Let $H$ be a infinite dimensional complex ...
4
votes
5answers
266 views

Possible flaw in “proof” that a sum of two compact operators is compact

If X and Y are Banach spaces, and $A: X \to Y$, $B: X \to Y$ are both compact operators, then $A + B$ is compact. A + B is compact if and only if for every bounded sequence $\lbrace x_n \rbrace$ ...
0
votes
0answers
40 views

Does the orthogonal projection theorem guarantees uniqueness of the projected space?

Given a Hilbert space $H$, and linear map $P:H \to H$ such that $P^2=P$ and for every $x\in H$ : $\|Px\| \le \|x\|$, there is a closed linear-subspace $M$ such that $P=P_M$, the projection on $M$. My ...
2
votes
1answer
58 views

Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
3
votes
1answer
39 views

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ...
1
vote
1answer
30 views

Does Hilbert–Schmidt theorem imply the space is separable?

The Hilbert–Schmidt theorem says a self-adjoint compact operator on a Hilbert space have a complete orthonormal set consisting of eigenvectors. Does that imply the space is separable?
0
votes
0answers
33 views

If $A$ is a compact diagonal operator, with diagonal $\{\alpha_n\}$, then $\lim_{n\to\infty}\alpha_n=0$.

Here is my question: If $A\in \mathscr{B}(\mathscr{H})$ is a diagonal operator with diagonal $\{\alpha_n\}$, show that if $A$ is compact, then $\lim_{n\to\infty}\alpha_n=0$. Here is what I have: I ...
2
votes
0answers
26 views

Why is the total time derivative of this partial space derivative zero?

A Lax pair for the Burgers equation $u_t+2 \, u \, u_x+ u_{xx} =0$ is, $$L = \partial_x +u \text{ and } M=-\partial_{xx} -2 \, u \, \partial_{x}$$ To get the resulting differential equation from the ...
2
votes
0answers
27 views

Compact operators on a Banach space $X$ are closed in the bounded operators on $X$. - Proof correction help

I am given a proof of the following statement (see below). Compact operators on a Banach space $X$ are closed in the bounded operators on $X$. I was told that there is an error in this proof - I ...
3
votes
1answer
35 views

Prove $Tx=(r_1x_1, r_2x_2, r_3x_3,…)$ is compact, $T:l^2\to l^2$, $r\in l^2$

Here is my question: Fix $r=(r_1,r_2,...)\in l^2$. Define $T:l^2\to l^2$ by $$Tx=(r_1x_1, r_2x_2, r_3x_3,...)$$ Prove that $T$ is compact. Here is what I have, input would be appreciated: Let ...
4
votes
1answer
66 views

Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
3
votes
2answers
33 views

Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq ...
0
votes
1answer
10 views

Let L be a bounded linear operator on a Hilbert space H. Verify the following relationships: $null(L^*)=null(LL^*)$

Just started to learn about linear operator theory, and trying to understand adjoint operator. Here's a conceptual problem, can someone help me to clarify? Thanks Let L be a bounded linear operator ...
0
votes
1answer
24 views

Spectral Measures: Commuting Operators

The questions are given below!! Theorem Given a measure space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. Denote ...
3
votes
1answer
53 views

How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

Without using Fourier transforms, how do I derive the spectral measure for $A=\frac{1}{i}\frac{d}{dt}$ on the domain $\mathcal{D}(A)$ consisting of absolutely continuous functions $f\in ...
1
vote
1answer
31 views

Why is this operator one-to-one

I am reading a textbook, and would like to ask a question about the proof. Here $S_p$ is the Schatten p class. My question is, in the proof, why is $A: X\to H$ is one-to-one? I actually don't ...
1
vote
1answer
59 views

Why is this operator self-adoint

We have that $\lambda, \overline{\lambda} \in \rho(T)$ and $\lambda \in \mathbb{C}$. Now, I want to show that a symmetric operator and closed operator $T: \operatorname{dom(T)} \rightarrow H$ must be ...
0
votes
2answers
15 views

What is an involutive operator

Please help me in understanding this: I have to find the eigen values of an involutive operator. So what exactly is an involutive operator? I mean I need one example for an involutive operator. ...
0
votes
0answers
36 views

Norm of an operator

Suppose $\{\xi_i\}_{i\in I}$ is an orthonormal system of Hilbert space $H$ and $T\in B(H)$. For each $i\in I$, let $\alpha_i$ be a scalar of modulus one such that $$|(T\xi_i,\xi_i)| = \alpha_i ...
0
votes
1answer
15 views

Nonclosable Operator: Example (Wikipedia)

The example here is taken from the wikipedia article: Discontinuous Linear Map Given the spaces of polynomials $X:=\mathcal{P}([0,1])$ and $Y:=\mathcal{P}([2,3])$. Their completions being ...
1
vote
1answer
38 views

Norm of a sequence

The following is a theorem that I have some difficulty at it. I do not know how the author shows that $\alpha \in \ell^1$. Please help me. Thanks in advance.
2
votes
2answers
22 views

A question about spectral measure

The following is a part of a theorem of Takesaki's Operator theory: Let $T$ be an positive operator. Suppose $T = \int_0^{\|T\|} \lambda \, de(\lambda)$ is the spectral measure of $T$. Also put ...
3
votes
1answer
36 views

Is it true that $\|A+PBP\|\le\|A+B\|$ for every projection $P$ and positive operators $A,B$?

Let A and B be positive operators on and let P be a projection. Is the inequality $$\|A+PBP\|\le\|A+B\|$$ true? Here $\|.\|$ stands for the operator norm.
0
votes
1answer
27 views

A question about finite-rank projection on Hilbert space

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, Can we verify that ...
1
vote
2answers
55 views

Bounded measurable functions

Suppose $X$ is a compact space and $B(X)$ denotes the bounded Borel measurable function space. Let $f\in B(X)$. There is a sequence of step functions $\{\phi_n\}$ such that $\phi_n\to f$ (point wise). ...
1
vote
1answer
58 views

Resolvent also self-adjoint operator

If I have a self-adjoint operator $U : \operatorname{dom}(U) \subset H \rightarrow H$ and $\lambda \in \rho(U)$, then I assume assume that it is correct that the operator $(U - \lambda I)^{-1} \in ...
2
votes
1answer
36 views

A question about spectral theorem

The following is a discussion about spectral theorem of Folland's Harmonic analysis page 18. Suppose $A$ is a unital commutative C*- subalgebra of $B(H)$ and $u,v\in H$. Put $\Sigma = \sigma(A)$ . ...
1
vote
2answers
36 views

Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...