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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Orthogonality of projections on a Hilbert space

Assume that $p$ and $q$ are (orthogonal) projections on Hilbert space $\mathcal{H}$. I want to prove: $pq=0$ iff $p+q\leq1$ I had the following in mind: Assume $pq=0$. Then $qp=0$, hence $p+q$ is a ...
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13 views

Image of bounded linear operator?

Let $x^\ast$ be a continuous linear functionals on $l_p$. Let $(e_i)_{i\in \Bbb N}$ be the standard basis of $l_p$. Consider $y=(y_i)_{i\in \Bbb N}$ the sequence defined by $y_i=x^\ast(e_i)$. Let ...
0
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1answer
10 views

Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
0
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1answer
20 views

Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
2
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1answer
28 views

Is this a bounded linear operator?

I have the following problem. Show that $y_n={1 \over \sqrt n}\int_0^1t^nx(t)dt$ is a bounded linear operator that maps $L_2[0,1]$ into $l_2$ with the usual norm on the respective spaces. My approach ...
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1answer
25 views

Reference for unbounded operators

I've run into some unbounded operators in my research and need to learn some of the theory of unbounded operators. Particularly I want a rigorous treatment that discusses symmetric operators, ...
0
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0answers
28 views

Need an operator with given properties [on hold]

Need an expression for an operator $U:f\to f^*$ satisfying the following properties: $U$ is linear $U$ depends only on the local properties of $f(x)$ in the neighbourhood of $x=0$ ...
2
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1answer
36 views

spectrum of convolution integral operator

Let $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$ operator $L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi}), h$ is continuous, periodic with period $2\pi$ and $h(x)=h(-x)$ on $ [ {-\pi},{\pi}] $. How can I ...
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0answers
8 views

Algebraic multiplicity of an eigenvalue for abstract operators

How does one define algebraic multiplicity of an eigenvalue for an abstract operator? (for a matrix the definition is clear). E.g. Consider $\partial_x^2$ on $H^2_{per}(0,1)$ then $\partial_x^2 ...
2
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2answers
40 views

Self-adjoint operator- domain unique?

I was wondering about the following: Let $T : dom(T) \subset H \rightarrow H$ be a self-adjoint operator, does this mean that the domain of $T$ is uniquely defined or is it possible to make the same ...
0
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1answer
14 views

The domain of the generator of a strongly continuous semi-group is dense

Let $A$ with domain $D(A)\subset X$, ($X$ Banach) be the generator of a strongly continuous semi-group $(S(t))_{t\geq 0}$. Then $D(A)$ is dense in X. I am not sure if this proof is correct. I know ...
0
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1answer
21 views

Using substituion rule for piecewise monotonic function

I am trying to read the ON THE EXISTENCE OF INVARIANT MEASURES FOR PIECEWISE MONOTONIC TRANSFORMATIONS paper from 1973. One has given a map $\tau : [0,1] \to [0,1]$ which is a piecwiese monotonic ...
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1answer
22 views

characterizing an operator with projection whose spectrum is contained in $\{-1,1\}$

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\sigma$ denote the spectrum. I want to show that if $\sigma (A)\subseteq \{-1,+1\}$ for $A\in \mathcal{A}$ then there is a projection $P$ such that ...
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0answers
21 views

States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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0answers
7 views

Pseudodifferential Operators are properly supported iff the symbol is properly supported

I am trying to proof the following statement: Pseudodifferential Operators are properly supported iff the symbol is properly supported. A Pseudodifferential Operator $A \in \Psi^m(X)$ ($\Psi^m$ ...
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2answers
53 views

Fock Space: KMS-State

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: ...
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2answers
39 views

Compact operator space is the greatest ideal of $B(H)$

Suppose $H$ is a separable infinite dimensional Hilbert space. Show that if $A\in B(H)$ is noncompact, then there exist two operators $B,C$ such that $BAC=1$. Clearly if $A$ is invertible it holds, ...
2
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1answer
89 views

Show that the operator $(x_n)_n\mapsto (\frac{x_n}{n}) $ is compact

I want to show that the following operator is compact: $$T:\mathbb \ell^p\rightarrow \mathbb \ell^p, \text{ }(x_n)_n\mapsto(\frac{x_n}{n})_n \text{ } 1\leq p<\infty$$ Its the first time that ...
2
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0answers
15 views

Does every closed, densely operator in a Banach space have an closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. ...
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1answer
39 views

Adjoint of Integral Operator in $L^p$

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Find the adjoint of $T$. I know how to this in the case $p=2$ as shown here. But in general $L^p$ is not an ...
0
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1answer
43 views

Møller Operators: Summary

Disclaimer This thread is meant as summary. For more informations see: SE blog: Answer own Question MSE meta: Answer own Question (The second especially reveals the opinion of the community!) ...
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1answer
23 views

Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
2
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39 views

Boundedness of linear operators in $L^p$

I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces: i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$. ii) There exists ...
0
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34 views

Hilbert- Schmidt class is an ideal

Definitions: 1 - An operator $y\in B(H)$ is said to be of trace class if $y$ is compact, and also $\sum|\alpha_n| <\infty$ where $\alpha_n \in \sigma(y)$ and $y$ has a representation $\sum ...
0
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1answer
31 views

weak convergence of $L^2$ implies weak convergence of $W_0^{1,2}$ (up to a subsequence)?

In the paper that I am reading, it says that if $\{u_n\}$ are bounded in $W_0^{1,2} (\Omega)$ (bounded $\Omega\subset \mathbb{R}^N$) and $u_n \rightharpoonup u$ weakly in $L^2 (\Omega)$, then there ...
0
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1answer
28 views

Differential operator a bounded operator or not?

Is the operator $T$ a bounded operator mapping $T: H^n([0,\pi]) \rightarrow H^{n-1}([0,\pi])$ ($H^n$ is the n-th Sobolev space with respect to $L^2$) or not? The operator itself is given by ...
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29 views

Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
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1answer
41 views

Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
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23 views

Finite dimensional operator space is dense in trace class space

To show that $F(H)$ (the space of finite dimensional operators on a Hilbert space $H$) is dense in $L^1(H)$ (the space of trace class operators), suppose that $x\in L^1(H)$. Without loss of generality ...
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1answer
17 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
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30 views

Trace class operator

Let $A\in B(H)$ and $\sum_{E}|\langle A e,e\rangle|< \infty$ for every orthonormal basis $E$. Show that $A$ is a trace class (means $\sum_E \langle |A|e,e\rangle < \infty$). I can not prove it. ...
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26 views

Compact operator and a sot convergent sequence of operators

The following is an exercise of Conway's operator theory: I proved all parts of this exercise except $\|KT_n\| \to 0$. I can easily prove $\|KT_n^*\|\to 0$, but do not have any idea to prove ...
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1answer
32 views

Continuous operator between Banach spaces, closed range

I have some problems proving the following: $T: X \rightarrow Y$ is a continuous, linear operator between Banach spaces. Prove that $T$ is surjective $\iff$ $T^* : Y^* \rightarrow X^*$ is injective ...
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Functional Analysis- Operators [closed]

Question: Consider the operator $A:L^2[0,1]\to\mathbb{R}$ defined by $(Af)x=f(x^\alpha)$. Determine the operator $A^*$.
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1answer
42 views

On Fredholm operator on Hilbert spaces

Let $u: H \to H'$ be a continuous linear operator and $H,H'$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite ...
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6answers
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Operator vs function

Could someone please explain the MATHEMATICAL difference between an operator and a function? I am not talking about these in terms of coding but rather the mathematical difference. Is operator also a ...
0
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1answer
21 views

Irridicible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

Let $A$ be an irreducible C*-subalgebra of $B(H)$ and $p$ be a nonzero projection in $B(H)$. Suppose $pAp=\Bbb C p$, show that $p$ is rank one. I do not have any idea about it. Please give me a ...
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Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
2
votes
1answer
39 views

Finite rank volterra operator

I am wondering when a Volterra integral operator $V_K:L_2(0,1)\to L_2(0,1)$ is a finite rank operator: $$V_Kf=\int_0^xK(x,y)f(y)dy$$ thanks in advance for your help
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0answers
26 views

$M_{n}(A)$ is an AF-algebra

If $A$ is a $C^*$-algebra that contains an increasing sequence $(A_{n})_{n=1}^{\infty}$ of finite-dimensional $C^*$-subalgebras such that $\cup‎_{n=1}^{\infty} A_{n}$ is dense in $A$, show that ...
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1answer
320 views

Matrix form of the differential operator $\sum_{k=1}^N x^k\frac{d^k}{dx^k}$

The following differential operator: $P(x,N)=\sum_{k=1}^N x^k\frac{d^k}{dx^k}$ is defined in $x\in\left[-1,+1\right]$. Is it possible to find a matrix form of this operator vs. $N$? Because it's ...
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Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
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3answers
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Why is $R-\lambda$ invertible for $|\lambda|<1$

I got the following question: Why is $R-\lambda$ invertible for $|\lambda|>1$ and not invertible for $|\lambda|\leq1$ ? R is the right shift operator on $\mathfrak{l^2}$
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2answers
48 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims that$$u = {\rm strong} - \lim_{\epsilon\to 0} ...
2
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1answer
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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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1answer
20 views

Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N $ and $N$ is finite, then $M$ is finite. I think it should say ...
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2answers
48 views

From continuous to bounded Borel functions

I know that we can extend the functional calculus of bounded self-adjoint operators to bounded Borel functions. I want to do the same for unbounded self-adjouint operators. Therefore assume that $T$ ...
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106 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
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2answers
36 views

Fock Space: Formal Adjoints

Problem Given a pre-Hilbert space $\mathcal{H}$. Consider unbounded operators: $$S,T:\mathcal{H}\to\mathcal{H}$$ Suppose they're formal adjoints: $$\langle ...
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24 views

Invariant subspace of bounded self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...