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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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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27 views

Properties of trace-class operators

Let $X$ be a separable Hilbert space (real or complex). Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, and suppose $B\in\mathcal{L}\left(X\right)$, which is of trace-class. ...
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2answers
36 views

Operator in Denominator

So I chanced upon this statement, and I'm not sure what is happening: $$ \left(c-\frac{1}{b}\frac{\partial}{\partial ...
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1answer
29 views

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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1answer
38 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. [duplicate]

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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54 views

Wave Operators: Preliminary [closed]

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

Stampacchia Problem

I need to solve this problem, but don't know how get that particular bound. Please, somebody can help me? Let $V$ a Hilbert space, $a : V\times V\rightarrow\mathbb{R}$ a bounded bilinear form, ...
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28 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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46 views

Spectral Measures: Constructions

Any constructions welcome!!! Given a Hilbert space $\mathcal{H}$. Regard spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ That are additive: ...
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124 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
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38 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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85 views

Spectral Measures: Stone's Formula

Given a Hilbert space. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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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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32 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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24 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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1answer
32 views

Spectral Measures: Multi Version (I)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad ...
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2answers
48 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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57 views

Resolvent: Decay Behavior

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote resolvent set: ...
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1answer
296 views

Positive unbounded operators

Let $T$ be an operator in $H$. We say self adjoint $T$ is positive iff $(\forall x\in H)\langle Tx,x\rangle \geq 0 $. As in the case of bounded operators, it is true that a self-adjoint operator $T$ ...
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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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51 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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384 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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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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60 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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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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2answers
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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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For a multiplication operator $M_f$ on $L^2$ with $f\geq 0$, is $SM_fS^{*}$ positive?

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
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1answer
357 views

If $0\leq A\leq B$ on Hilbert space and $A^{-1}$ exists, show that $A^{-1}\geq B^{-1}$ [duplicate]

Does anyone know how to show this? Let $H$ be a Hilbert space and $A$, $B$ bounded positive operators defined on $H$ such that $A^{-1}: H \rightarrow H$ exists and hence bounded and $A \leq B$. ...
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681 views

For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
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1answer
140 views

Does inversion reverse order for positive elements in a unital C* algebra?

Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related ...
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1answer
79 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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35 views

Is range a of a generator of a strongly continuous semi group in doman of the generator?

Let $X$ be a banach space and $A:D(A)\rightarrow X$ be a generator of a infinte seminal generator of a $C_0$ semi group $\{S(t)\}_{t\geq 0}$. In this case is it possible that ...
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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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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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1answer
61 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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1answer
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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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Spectral Measures: Helffer-Sjöstrand

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a function: $$f\in\mathcal{C}^\infty_0(\mathbb{R}):\quad ...
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1answer
11 views

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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1answer
41 views

Mourre Adjoint: Bounded Maps (II)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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1answer
49 views

Mourre Adjoint: Bounded Maps (I)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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1answer
49 views

Mourre Adjoint: Bounded Maps (III)

I will provide an answer later... Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: ...
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Compute the spectrum of the integral operator $K:L^2([0,1]) \to L^2([0,1])$ defined as $(Kx)(t) = \int_0^t x(s) ds$

Let $K:L^2([0,1])\rightarrow L^2([0,1])$ be the linear operator defined by $$(Kx)(t)=\int_0^tx(s)ds, \quad x \in L^2([0,1]).$$ Now I have to compute the spectrum, but I don't have any idea how to do ...
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2answers
32 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 ...
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23 views

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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1answer
29 views

Prob. 10, Sec. 3.10 in Kreyszig's functional analysis book: Every isometric linear operator on a finite-dimensional inner product space is unitary? [duplicate]

Let $X$ be an inner product space such that $\dim X < \infty$, and let $T \colon X \to X$ be an isometric linear operator. Since $\dim X < \infty$, $X$ is complete and thus a Hilbert space; ...
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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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1answer
30 views

x-momentum operator $p_x$ expressed as multiple of Translation operator

On this page https://en.wikipedia.org/wiki/Rotation_operator_%28quantum_mechanics%29 under "The translation operator," they use Taylor expansion. As part of that proof they state $p_x = ih * ...
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0answers
42 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? ...
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1answer
148 views

An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
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Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...