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

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

Let $V$ a normed space of infinite dimension and let $W\neq \emptyset$ a normed space. Prove that $Hom(V,W)\neq L(V,W)$.
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1answer
26 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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0answers
13 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
57 views
+50

Modulus: Invariant Domain

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense domains: ...
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2answers
38 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
55 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
286 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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1answer
5 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: ...
2
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2answers
83 views

Spectral Measures: Scale Spaces (II)

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: ...
0
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1answer
29 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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1answer
18 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: ...
0
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1answer
13 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: ...
2
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0answers
27 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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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"
4
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1answer
41 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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0answers
20 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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1answer
381 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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0answers
33 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
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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0answers
25 views

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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3answers
55 views

Selfadjoint Operators: Sesquilinear Form (II)

Given a Hilbert space $\mathcal{H}$. Consider a positive form: $$s:\mathcal{D}\to\mathcal{H}:\quad s(\varphi,\varphi)\geq0$$ Introduce its form space: ...
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2answers
33 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

Normal Operators: Transform (III)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$N=Z\sqrt{1-Z^*Z}^{-1}$$ Especially one had: ...
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2answers
344 views

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 ...
2
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1answer
350 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 $A \leq B$. ...
4
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1answer
676 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
139 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 ...
5
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1answer
41 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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2answers
33 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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1answer
42 views

Polar Decomposition: Ranges

This is just a note. Given Hilbert spaces $\mathcal{H}$, $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{K}:\quad A=A^{**}$$ Construct its modulus: ...
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1answer
73 views

Spectral Measures: Polar Decomposition

Isometric Equality Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^{**}$$ It gives rise to operators: ...
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1answer
53 views

Spectral Measures: Reducibility

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: ...
4
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3answers
75 views

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
52 views

Wave Operators: Functional Calculus

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_0:\mathcal{D}(H_0)\to\mathcal{H}_0:\quad H_0=H_0^*$$ $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and a bounded ...
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0answers
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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)\\ ...
1
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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?
2
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1answer
39 views

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 ...
3
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1answer
98 views

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 ...
0
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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: ...
0
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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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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 ...
4
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1answer
66 views

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 ...
0
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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 ...
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0answers
21 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
28 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; ...
2
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1answer
28 views

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 ...
0
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1answer
24 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 * ...