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.

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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 $\hat\rho$, ...
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2answers
35 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 $x \in C^0([0,1],\mathbb R)$.
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1answer
33 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 ...
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2answers
40 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: ...
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22 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 ...
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83 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 ...
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1answer
55 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 $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
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1answer
83 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 ...
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29 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 ...
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5answers
275 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 $\lbrace x_n \rbrace$ ...
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42 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 ...
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1answer
60 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 ...
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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 ...
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1answer
34 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?
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37 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 ...
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0answers
27 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 ...
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0answers
29 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 ...
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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 ...
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1answer
81 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 ...
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2answers
40 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 ...
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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 ...
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25 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 ...
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60 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 ...
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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 ...
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1answer
64 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 ...
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2answers
19 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. ...
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39 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 ...
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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 ...
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1answer
39 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.
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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 ...
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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.
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30 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 ...
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61 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). ...
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1answer
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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 ...
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1answer
37 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 $\Sigma = \sigma(A)$ . ...
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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 $K$ ...
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1answer
55 views

Limit-circle and limit-point

Imagine that we have a second-order Sturm-Liouville problem on an interval $(a,b)$. If every solution is square integrable, then our operator is called limit-circle and if there is at least one ...
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1answer
35 views

Show that $\lbrace S_n x \rbrace$ converges for a particular recursively-defined sequence of operators $S_n$

$H$ is a Hilbert space, $M$ is a self-adjoint bounded linear operator on $H$ with $M \leq I$, and $S_0 = 0$; $S_{n+1} = (1/2)(M + S^2_n)$ for $n = 0, 1, 2, ...$. For all $n$, both $S_n$ and $S_n - ...
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Rotation in configuration space.

Let $R_\psi$ be the rotation in configuration space around a vector $\bf{e}_\psi$ for an angle $\psi$. How is that the space rotation in configuration space have: ...
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1answer
37 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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1answer
49 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
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1answer
34 views

Correspondence between bounded sesquilinear forms and bounded linear operators

Let $H,K$ are Hilbert spaces, I want to show there is an isometric linear correspondence between bounded sesquilinear forms $S(H,K)$ and bounded linear operators $B(H,K)$. ( $\Phi: B(H,K)\to S(H,K)$ ...
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1answer
29 views

Positivelinear operator on $L^p$-spaces

Suppose $1<p<\infty$. A linear operator $T \colon L^p(\Omega)\to L^p(\Omega)$ is positive if $f \geq 0$ imply $T(f)\geq 0$ (where $\Omega$ is a measure space). 1) Does there exist a positive ...
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1answer
35 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
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61 views

A question about sublinear functionals

Could you please give me hints may leads to prove the following: Let $X$ be a real vector space, $\,p_1,p_2:X\to\mathbb R\,$ be two sublinear functionals, and $\,f:X\to\mathbb R\,$ be a linear ...
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61 views

About the adjoint operator and weak operator topology.

Let $X,Y$ be Banach spaces. Let $\lbrace{S_n\rbrace}\subset\mathcal{L}(X,Y)$, and $T\in\mathcal{L}(X,Y)$, such that $S_n\xrightarrow[n\to\infty]{WOT}T$, that is: $$\langle ...
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0answers
10 views

Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
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Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$.

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$. I've been stuck on this for a while and don't really know where to start.
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1answer
45 views

Showing an operator is essentially self-adjoint

I have a question about checking if an operator is essentially self-adjoint. Given the operator $$H=-\frac{1}{2}\partial^2_{r}-\frac{1}{r}\partial_r$$ with domain $C^{\infty}_0((0,\infty))$ (i.e. ...
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1answer
31 views

Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...