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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Action of projections

Suppose we have a projection $p$ on a Hilbert space $\mathfrak{H}$. Is the following true: There exists an set $V\subset\mathfrak{H}$ such that $p(x)=x$ if $x\in V$ and zero else? I asked because I ...
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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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1answer
15 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 ...
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2answers
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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 ...
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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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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 $x=0$ ...
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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 ...
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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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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 ...
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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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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, ...
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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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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 ...
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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
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 ...
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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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44 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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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)| ...
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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 ...
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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 ...
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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 ...
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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, ...
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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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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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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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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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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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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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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 ...
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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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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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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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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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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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$M_{n}(A)$ 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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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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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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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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1answer
30 views

Prove of some properties about unitary operators [closed]

Let $X$ be a hilbert space and $T\in L(X)$ be an unitary operator. Show (1) $\sigma(T)\subset\{\lambda \in \mathbb C:|\lambda|=1\}$ (2) for $\lambda \in \mathbb C$ with $|\lambda|\neq1$ holds: ...
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What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
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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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1answer
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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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38 views

Nature of the infinite differential sum operator?

Consider the operator $$ Hf = f + f' + f'' +\cdots = \sum_{i=0}^\infty \left[ \frac{d^i f}{dx^i}\right] $$ I am trying to determine what $ Hf $ is entirely in terms of $f$. I note the following ...
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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 ...
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1answer
31 views

Invariant subspace of 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 ...
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Ask for the name of a condition on commutator of two operators

Let $T, S$ be two bounded linear operators on a Hilbert space. I wonder whether there is a standard way referring the following condition: $$ \text{The commutator $[T, S]$ is in the Hilbert-Schmidt ...
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Group operator f where the notion of $f^{-1}$ is well defined and easily computable

I am reading a small part of text, related to algorithms: We may state the problem of range queries in the following way: a range query $q_f(A,i,j)$ on an array $A=[a_1,a_2,..,a_n]$ of n ...
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1answer
43 views

Von Neumann algebra generated by a subalgebra

Let A be a C*-algebra of operators on a Hilbert space H. Show that if $A\subset K(H)$, then $\{A'\cap K(H)\}'\cap K(H) = A$ I do not have any idea about it. Please give me a hint. Thanks.
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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 ...