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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If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint. I'm lost on this problem. I don't know how to even start this. Any solutions or hints would be ...
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Wave Operators: Adjoint

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
29 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint.

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
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Wave Operators: Cook

Problem Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote for shorthand: ...
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20 views

Show that an operator of rank $n$ can have at most $n$ nonzero eigenvalues.

Show that an operator of rank $n$ can have at most $n$ nonzero eigenvalues. I'm not sure how to proceed. I think induction will be the best. Any solutions or hints are greatly appreciated.
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Hamiltonian: Compactness

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the resolvent: $$R(z):=(z-H)^{-1}\in\mathcal{B}(\mathcal{H})$$ Denote compact ...
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27 views

Singular Spectrum: Criterion

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: ...
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12 views

Spectral Measures: Permutability

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E^{(\prime)}:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote their operators by: ...
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33 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
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1answer
49 views

prove the continuity of $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$ [duplicate]

Let $\phi\in C[0,1]$ and let $T_\phi~:C[0,1]\rightarrow\mathbb{R}$, defined as $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$. How can i prove that it's a continuous operator?
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1answer
31 views

Prove that $Hom(V,W)\neq L(V,W)$

Let $V$ a normed space of infinite dimension and let $W\neq 0$ a normed space. Prove that $Hom(V,W)\neq L(V,W)$.
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1answer
19 views

how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
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2answers
46 views

Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous

Let $k:[0,1]\times[0,1]\rightarrow \mathbb{R}$ continuous and $K:C[0,1]\rightarrow C[0,1]$, given by $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$. Prove that $K$ is continuous. I try to see continuity in $0$, ...
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1answer
7 views

Spectral Measures: Scale Spaces (V)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
16 views

Spectral Measures: Scale Spaces (IV)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
20 views

Spectral Measures: Scale Spaces (III)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
34 views

Existence of certain idempotents

Suppose $T$ is an idempotent (that is $T^2=T$) of infinite rank and co-rank on a separable Hilbert space. Can we find an idempotent $S$ such that $\overline{TS(H)}=(Id-S)(H)$?
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30 views

Spectral Measures: Scale Spaces (I)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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2answers
26 views

linear operator on normed spaces

Let $V$ and $W$, normed spaces and $T:V \to W$ a linear operator. How to prove that: "if $T$ is continuous in $0$, so, $\forall A \subset V$ bounded, $T(A)$ is also bounded"
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35 views

Sot convergence of a sequence of operators implies uniform convergence

Let $H$ be a Hilbert Space. Let $\{A_n\}$ be a sequence of bounded operators in $H$, and $A\in B(H)$. If $\|A_nf - Af\|\to 0$ uniformly for $f\in H_{\|.\|=1}\ $, prove that $\|A_n - A\|\to 0$. ...
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1answer
43 views

Range of normal operator and its adjoint are equal

On Wikipedia it is written that bounded normal operator in Hilbert space has the same range and kernel as its adjoint. I've been able to show equality of kernels and closures of ranges: ...
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22 views

Integral kernel of operators from the functional calculus theorem

Let $T$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ and take any bounded Borel function $F:[0,\infty)\rightarrow\mathbb{C}$. Does $F(T)$ -defined by the functional calculus ...
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is this conclusion true or false?

Let $\mathcal{A}$ be a factor Von Neumann algebra and $\Phi$ is a map on $\mathcal{A}$ which is injective and surjective and $\Phi(0)=0$. If $A, B, C \in \mathcal{A}$ and ...
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56 views

Operator theory curiosity

I'm not an expert in operator theory... but i was wandering if there's some practical applications. For example (the first one i came up with) compared to normal calculus techniques that usually the ...
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2answers
34 views

Kernel of a bounded linear operator on a normed linear space need not be closed or open?

How should be the kernel of a bounded linear operator on a normed linear space as a set? Kernel of a bounded linear operator on a normed linear space need to be closed or open? Or it need not be ...
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1answer
43 views

Definition of resolvent set

I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
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13 views

Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...
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3answers
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Is this operator $A = \pmatrix{1&1\\0&1}$ self-adjoint?

Is this operator $$A = \pmatrix{1&1\\0&1}$$ self-adjoint? I think not, because $$\pmatrix{1&1\\0&1}^T\neq A$$ where $T$ is the transposition of the matrix. What do you all think?
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1answer
53 views

Does $A$ and $(A+I)^{-1}$ commute for positive operator $A$?

Suppose that $A$ is a bounded positive operator ($A \geqslant 0$) on some Hilbert space. Can I say that $A$ and $(A+I)^{-1}$ commute?
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Example: Operator with empty spectrum

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your ...
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1answer
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Prove that $\|I-S\|=\sup_{\|x\|=1} |((I-S)x,x)|\leq c$.

Let $S$ a linear, self-adjoint, bounded and positive operator. In a document I'm reading, it says that since $0\leq I-S\leq cI$ with $c<1$ then $$\|I-S\|=\sup_{\|x\|=1} |((I-S)x,x)|\leq ...
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Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded? [closed]

Define operator $S$ and $M$ on $\ell^2$ by $(SX)_n = \begin{cases} 0 & n = 0 \\ x_{n - 1} & n > 1 \end{cases}$ $(Mx)_n = \dfrac{1}{n + 1} x_n,\qquad n \ge 0$ Why does $M$ have a limit ...
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Domain of closed unbounded operator

Let $A$, $B$ be two closed unbounded operators such that: (1) there exists dense subspace $\mathcal{D}$ of $Dom(B)$ which is contained in $Dom(A)$, (2) for every $\psi \in\mathcal{D}$ it holds $$ ...
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2answers
30 views

Some conditions on self-adjoint operator.

I have a bounded, invertible and positive operator on an $N-$dimensional Hilbert space $V$. I want prove or disprove that it is also self-adjoint. I would like to read an answer with some approaches ...
2
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1answer
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Existence of the continuous spectrum of a possibly-unbounded, linear self-adjoint operator on a complex Hilbert space

Let $\mathbf{A}$ be a possibly-unbounded, linear self-adjoint operator on an infinte-dimensional, complex separable Hilbert space $\mathcal{H}$, and suppose we know the matrix elements $\langle ...
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38 views

Difference between continuous and essential spectrum, examples?

Can anyone help me to understand the definitions of the continuous and essential spectra in simple terms and point out the difference on examples where the two definitions coincide and where don't? ...
2
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1answer
28 views

Mathematical Operator to flip a vector

Is there a mathematical operator which flips a vector from left to right (or up to down). Say \begin{align} a = [1~ 2~ 3]\quad\text{and}\quad b = [3~ 2~ 1] \end{align} I'd like to have \begin{align} a ...
3
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0answers
43 views

Evaluate the norm of a linear operator

Can someone help me on this question. I want to compute the norm of the following operator $$l:\mathbb R^N\longrightarrow \mathcal M_n(\mathbb R); (x_1,\cdots,x_N)\mapsto (x_i - x_j)_{1\leq i,j\leq ...
5
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1answer
65 views

Equivalent formulations: pure contraction

I want to prove the following equivalence: let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. TFAE: $\|Tx\|<\|x\|$ for each $x\in H\setminus\{0\}$ $\|T\|\leq1$ and ...
2
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0answers
121 views

Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

Let $H$ be a separable, infinite-dimensional Hilbert space, and $B(H) = \{T : H \to H, T \space \text {is non-bounded and linear operator} \}$. We say An operator $T \in B(H)$ is chaotic if $T$ is ...
3
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1answer
40 views

Compact linear operator

Today in lecture we were told that for a linear compact operator $T$ on an infinite-dimensional Hilbert space with infinite-dimensional range, we have that $\ker(T)^{\perp}$ is infinite-dimensional, ...
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1answer
14 views

Range of operator always closed. Mistake in argument

Let $A \in L(X,Y)$ be a linear operator between Hilbert spaces and the operator $$\hat{A}: \ker(A)^{\perp} \rightarrow \operatorname{ran}(A)$$ is a restriction of $A$ which is bijective. Now ...
2
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1answer
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Can a sequence of von Neumann algebras determine a maximal directed set of subalgebras?

Can a von Neumann algebra $A$ have an infinite sequence $A_0 \subset A_1 \subset A_2 \subset ...$ of sub-vN-algebras such that every other sub-vN-algebra $B \subseteq A$ satisfies, for some $n \geq 0$ ...
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0answers
8 views

How can I apply the Magnus expansion here?

Suppose I have two operators $A(t),V(t)$. There is also a parameter $t \in [0, \infty]$. Moreover I have a continuous function $f(t)$ which satisfies $A(s)f(t)=0$ for all $s \in [0,\infty]$. How can I ...
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0answers
9 views

Exponential operator on function; can it be simplified?

Suppose I have two operators $A(t),V(t)$. There is also a parameter $t \in [0, \infty]$. Moreover I have a continuous function $f(t)$ which satisfies $A(s)f(t)=0$ for all $s \in [0,\infty]$. How can I ...
0
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1answer
36 views

How to find the image of an arbitrary element under this operator?

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T \colon H \to H$ such that $T e_n = e_{n+1}$ for $n = 1, 2, ...
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1answer
23 views

Spectral Measures: Special Spectrum

Problem Given a Hilbert space $\mathcal{H}$. Denote eigenvalues by: $$\sigma_0(N):=\{\lambda\in\mathbb{C}:\mathcal{N}(\lambda-N)\neq(0)\}$$ Then arbitrary sets admit: ...
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1answer
13 views

Reducing Spaces: Decompostion

This thread is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Regard a decomposition: ...
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1answer
28 views

Spectral Measures: Multi Version (III)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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0answers
36 views

Formal decomposition of Hamiltonian into $A A^\ast$

Let $H = -\frac{d^2}{dx^2} + q$. Letting aside consideration of domains, I want to show that $H$ can be formally written as $H = A A^\ast$, where $A = -\frac{d}{dx} + \phi$ with some $\phi$ under the ...