Questions tagged [operator-theory]

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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Unilateral shift in strong and weak operator topologies

Hi there I am trying to understand how the unilateral shift operator converges in the strong and weak topologies. Here is the question: Suppose $S$ is a unilateral shift on $ℓ^2$ or $H^2$. Does ...
2
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2answers
52 views

Eigenvalues and Eigenvectors of Selfadjoint Operators

I am trying to show the following: Let $H$ be a Hilbert space. Suppose that $\|Tx\| = \|T\|$ for some unit vector $x \in H$ and for some bounded self-adjoint operator T on H. Then x is an eigenvector ...
2
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1answer
65 views

Is the differentiation operator normal? [on hold]

In the space of polynomials (degree not higher than n) , the scalar product is given by the formula: $$(f,g)=\int_{0}^{1}f(x)g(x)dx$$ is the differentiation operator normal?
5
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1answer
27 views

Operator norm of semigroup operator

Let $P_{t}$ be a self-adjoint operator such that $P_{t+s}=P_{t}P_{s}$. I want to show that $$\|P_{t}\|_{1\to \infty}\leq \|P_{t/2}\|_{1\to 2}\|P_{t/2}\|_{2\to \infty}.$$ For that, I am trying to ...
3
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21 views

The operator $a_n\frac{d^{2n}}{dx^{2n}}$ defined on $H^{n}_0(I)\cap H^{2n}(I)$ is closed.

Let $I \subseteq \mathbb{R}$ be an open interval with $I=(-a,a)$ for some $a>0$ or $I= \mathbb{R}$ or $I=(0, \infty)$, and let $n \in \mathbb{N}$. Consider the operator $T:H^{n}_0(I)\cap H^{2n}(I) \...
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1answer
17 views

Dimension of kernel of compact operator

Let $A$ be an invertible operator, and let $K$ be a compact operator in a Banach space. Prove that: a) $\dim(\ker(A+K)) \lt \infty$ b) $\operatorname{codim}(\operatorname{Im}(A+K)) \lt \...
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74 views

bounded below operator/ Kato-Rellich

I have shown that for a densely defined, self adjoint Operator D between Hilbert Spaces H, it holds $\inf \sigma(D)\geq c$ Is this enough to conclude, that $D$ is bounded below by c, i.e. $ < x, Dx&...
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Can't understand the definition that a kernel function is positive-definite. What is the type of its arguments?

I can't understand the solution to this example problem: Let $K_1:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ be an arbitrary kernel function. Prove that $K_2(x,y)=a\cdot K_1^2(x,y)+b$, ...
2
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1answer
28 views

Is Operator a Compact Operator?

So I have this linear operator: $$ A : C_{[0,1]} \rightarrow C_{[0,1]}: (Ax)(t) = \int_{0}^{1}(ts + t^2s^2)x(s)ds $$ And I need to check if it is a compact one. According to Arzela-Ascoli theorem, I ...
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1answer
33 views

Spectrum of inverse is contained

Let $A$ be a positive, self-adjoint operator in a Hilbert space such that $$\|A\|\leq 1/2.$$ Let $A^{-1}$ be its inverse (we suppose it exists). Now I want to show that $$\operatorname{spec}\left(A^{-...
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Is the failure of $\mathcal{B}(H)\simeq H\otimes H^*$ in infinite dimensions the reason for non-normal states in quantum information?

In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of ...
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243 views

How to justify solving $f(x+1) + f(x) = g(x)$ using this spectral-like method?

Let's say that I want to find solutions $f\in C(\Bbb R)$ to the equation $$ f(x+1) + f(x) = g(x) $$ for some $g\in C(\Bbb R)$. I can write $f(x+1) = (Tf)(x)$ where $T$ is the right shift operator ...
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Inclusion of the spectrum of two differential operators defined on $L^2[-a,a]$ and $L^2[0, \infty)$

Let $Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$ with $a_j \in \mathbb{C}$. Consider the differential operators $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ and $T_\infty: D(T_\infty)\subseteq L^2[0,\...
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1answer
34 views

Operatornorm of self-adjoint operator

Let $A$ be a self-adjoint, positive definite operator in $H$ with $\inf\, \text{spec}(A)\geq 1$. I want to show that it is equivalent to the conditions that $A$ is invertible and $||A^{-1}||\leq1$. ...
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1answer
192 views

Prove that a kernel operator has no eigenvalues

Good evening! I'm just popping here for a quick question. I'm just starting to work on kernel operators, from $L^2(\mathbb{R}_+)$ to itself, ie: $f \mapsto \left(x \mapsto \displaystyle\int_{\mathbb{...
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3answers
35 views

Showing $[A,(A^\dagger)^n]=n(A^\dagger)^{n-1}$

Given the operators $$A=\frac{1}{\sqrt{2}}\left(x+\frac{d}{dx}\right)\text{ and } A^\dagger=\frac{1}{\sqrt{2}}\left(x-\frac{d}{dx}\right)$$ and the commutator $[A,B]=AB-BA$ for Operators $A,B$,...
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36 views

Operator core of Dirichlet Laplacian

I am reading Reed-Simon Vol. 4, page 274. We have the following situation: Let $\Omega \subset \mathbb R^m$ be an $m$-dimensional cube and consider the Dirichlet-Laplacian $-\Delta_D=\Delta_D^\Omega$ ...
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Find $\alpha$ for which operator is compact - is my answer correct

Let $H$ be Hilbert space with basis $\phi_k$. We define the linear operator $T(\phi_k) = \frac{1}{(k+1)^\alpha}\phi_{k+1}$ Find for which values of $\alpha$ is $T$ compact. What I did: I will ...
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1answer
22 views

Reflection Operators and Self-Adjoint

Let R be a bounded operator on a Hilbert Space, H. I am trying to show that if X is a closed subspace of H such that \begin{align*} x + Rx \in X \ \ \ \text{and} \ \ \ x - Rx \in X^{\perp} \end{...
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If $K$ and $F$ are monotone, when is $I+KF$ monotone?

It is known that if $K$ and $F$ are monotone, that $I+KF$ may not be monotone. For example, if $F(x,y)=(x+y, y-x) $ and $K(u,v)=(u+2v, v-2u)$ then $F$ and $K$ are monotone. However, $I+KF$ is not ...
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35 views

Restriction of a differential operator on $L^2(\mathbb{R})$

I'm reading a proof of a paper and I don't understand an argument of the proof. That argument is the following: Let $A: D(A) \subseteq L^2(\mathbb{R}) \to L^2(\mathbb{R}) $ be a constant coefficients ...
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0answers
34 views

1.4.9 ‎Theorem ‎of‎ ‎Murphy's ‎book

The relevant theorem: Let $u$ be a compact operator on a Banach space $X$, and let $\lambda\in\mathbb{C}\setminus \{0\}$. $(1)$ The operator $u - \lambda$ is a Fredholm of index cero. $(2)...
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0answers
19 views

What is meant by Atomic Spectrum of operator?

I was going through some properties of linear operators. I came across the word "purely atomic spectrum". Can anyone help me in understanding its meaning?
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1answer
24 views

Positive Operators and Invertibility

Let T be a bounded operator on a Hilbert Space H. I am trying to show that $T + T^* \geq 0$ iff $T + I$ is invertible and $\left\vert\left\vert{(T - I)(T + I)^{-1}}\right\vert\right\vert \leq 1$. I ...
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0answers
8 views

definition of the partial trace and its complete positivity

I know the definition of the partial trace by its Kraus decomposition (which converges in trace norm to some trace class operator) $$ \mathrm{tr}_2(A) = \sum_k ( \mathbb{1} \otimes \langle f_k \rvert) ...
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1answer
729 views

Numerical (Second) Derivative of Time Series Data

First and second order derivatives are often used in chromatography to detect hidden peaks. The time series data consists of Instrumental Response vs. Time at very short time intervals (250 Hz). I ...
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1answer
25 views

Characterization of the generator of a measurable contraction semigroup

Let $E$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a contraction$^1$ semigroup on $E$. Assume $$[0,\infty)\to E\;,\;\;\;t\mapsto T(t)x\tag1$$ is Borel measurable for all $x\in E$ and hence $...
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1answer
33 views

Simple eigenvalue

A linear operator $T$ on a separable Hilbert space $H$ is said to be a weighted shift operator if there is some orthogonal basis $\{e_n\}_n$ and weight sequence $\{w_n\}_n$ such that $$Te_n=w_n e_{n+1}...
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0answers
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If $T-\lambda I$ is a Fredholm operator and $\lambda \in \sigma(T)$, then $\lambda \in \sigma_p(T)$ or $\overline{\lambda} \in \sigma_p(T^*)$?

Let $H$ be a Hilbert space and let $T: D(T) \subseteq H \to H$ be a densely defined operator. Is the following statement true? If $T-\lambda I$ is a Fredholm operator (an operator is Fredholm if its ...
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Can we show $\frac1h\int_t^{t+h}T(s)x\:{\rm d}s\xrightarrow{h\to0+}T(t)x$ for a measurable semigroup (with possibly discontinuous orbits)?

Let $E$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a semigroup on $E$. Assume $$[0,\infty)\to E\;,\;\;\;t\mapsto T(t)x\tag1$$ is Borel measurable for all $x\in E$ and $$\forall t>0:\sup_{...
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1answer
28 views

If $T(t)$ is a semigroup on $E$ and $B$ is a subspace of $E$ such that $\left\|T(h)x-x\right\|→0$ for all $x\in B$, is $T(t)$ locally bounded on $B$?

Let $E$ be a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ be a semigroup on $E$ and $B$ be a closed subspace of $E$ such that $$\left\|T(h)x-x\right\|_E\xrightarrow{h\to0+}0\;\;\;\text{for all }x\in B.\...
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40 views

If $T(t)$ is a semigroup and $x$ satisfies $\left\|T(h)x-x\right\|\to0$ as $h\to0$, are we able to conclude that $t\mapsto T(t)x$ is continous?

Let $E$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a semigroup on $E$. Definition: If $x\in E$, then $(T(t))_{t\ge0}$ is called strongly continuous at $x$ iff $$[0,\infty)\to E\;,...
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If $A$ is the generator of a semigroup $T$, are we able to show $\left\|λ(λ-A)^{-1}x-x\right\|\to0$ for all $x$ at which $T$ is strongly continuous?

Let $E$ be a $\mathbb R$-Banach space and $A$ be a dissipative linear operator on $E$. Assume $\lambda-A$ is surjective for all $\lambda>0$. Then we are able to show that $$\left\|\lambda(\lambda-A)...
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0answers
37 views

If $A$ is the full generator of a measurable contraction semigroup, is $\{(f,g)∈\overline A:g∈\overline{\mathcal D(A)}\}$ the conventional generator?

Let $L$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a conraction semigroup on $L$. Assume $$[0,\infty)\to L\;,\;\;\;t\mapsto T(t)f$$ is Borel measurable for all $f\in L$ and let $$A:=\left\{(...
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1answer
25 views

Spectral theorem on simple integral operator

Let $K(x): \mathbb R^d \to \mathbb R^d$, $K \in L^1(\mathbb R^d)$ such that $|K(x)| < M$ We define the integral operator $T$ as such: $\displaystyle(Tf)(x) = \int_{\mathbb R^d}f(y)K(|x-y|)dy$ ...
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1answer
295 views

Are all cyclic representations irreducible?

I know that for a representation $\pi$ of a $^*$-algebra $\mathcal{A}$ on a Hilbert Space $\mathcal{H}$, if $\pi$ is irreducible then it is cyclic. Is the reverse implication also valid - i.e. is ...
2
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1answer
27 views

Quotient of pre-hilbert space

Let $H$ be a pre-hilbert space and let $F$ be a complete subspace of $H$. Show that $H/F$ is a pre-hilbert space. My attemp: Consider the following theorem: $\it{Theorem}$: Let $H$ a pre-hilbert ...
2
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1answer
52 views

Showing that the operator is compact

Let $H$ separable Hilbert space and $(e_n)_n$ a orthogonal basis of $H$. Let $T$ a Hilbert-Schimidt operator and $a_n = T^*(e_n)$ where $T^* $ is Hermitian adjoint. I proved that $T(x)= \sum\langle x,...
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2answers
18 views

How can I show that this set is a closed subspace

Let $E:= \{(x_n) \in l^{2}(\mathbb N ) \mid x_{2k}=x_{2k+1}\}$ How can I show that $E$ is a closed subvector space of $l^{2}(\mathbb N )$ ? I tried to write $E$ as the kernel of a continuous linear ...
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1answer
20 views

Green function, determine $D(k)$ in $\,\,\,\,\, -(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}_{\rho}$

Given $g^{\mu\nu}=diag(1,-1,-1,-1)$ and $\delta^{\mu}_{\rho}$ the Kronecker delta. I'm in the fourier space: $$-(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}...
1
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1answer
787 views

Frechet/Gateaux differentiability of an integral operator $L^2 \rightarrow R$

Let $f: R \rightarrow R$ be a continuously differentiable function on the real numbers (if needed also infinitely many often differentiable). Define the Operator $F : L^2([0,1]) \rightarrow R$ for $x ...
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0answers
24 views

Representing linear operator with matrix

I've came across the following question in a book that I read about Hilbert spaces and linear operators - Where exercise $1$ is - So the book has answers, and for 13a it is - what I don't ...
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0answers
17 views

Equivalent definition for approximate point spectrum

Theorem Let $A$ be an operator on a Hilbert space $H$. The approximate spectrum denoted by $\sqcap (A).$ The following are equivalent: 1) $\lambda\in \sqcap (A).$ 2) There exists a sequence $\{f_n\}...
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0answers
44 views

Book recommendation request on Spectral theory

Can someone please recommend to me a text that deals on spectral theory from the scratch covering the parts of a spectrum (approximate, point and compression) explicitly. Theorems and properties. ...
1
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0answers
26 views

Show an inclusion for the range of a multi-valued dissipative operator

Let $Z$ be a $\mathbb R$-Banach space and $C$ be a multi-valued dissipative linear operator on $Z$, i.e. $C$ is a subspace of $Z\times Z$ with $$\forall\lambda>0:\forall(z,z')\in C:\left\|\lambda z-...
0
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0answers
40 views

extension of a GNS representation

There is a conclusion:Suppose $J$ is an ideal of a $C^*$ algebra $A$,if $J$ has a tracial state,then there is a unique extension of the GNS representation $\pi_{\tau}:J \to B(H_{\tau})$to a *-...
1
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1answer
25 views

extension of a non-degenerate representation of a $C^*$ algebra

The following statement is from Blackdar's book: If $J$ is a closed ideal of a $C^*$ algebra $A$,$\rho$ is a non-degenerate representation of $J$ on an $H$,and $(h_{\lambda})$ is an approximate unit ...
2
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1answer
33 views

Inequality in von Neumann algebras

U. Krengel. Ergodic Theorems. 1985. Page 278. Lemma 1.13: Let $U$ be a von Neumann algebra. If $p\in U$ is a projection and $a,b\in U$ satisfy $0\le a\le b\le 1$, then $||ap||\le \sqrt {||ap||}$. ...
2
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0answers
19 views

Show that this multi-valued operator is surjective (Theorem 1.6.9 of Ethier and Kurtz)

Let $L_n,L$ be $\mathbb R$-Banach spaces for $n\in\mathbb N$, $A_n\subseteq L_n\times L_n$ and $A\subseteq L\times L$ be linear and dissipative with $\mathcal R(\lambda-A_n)=L_n$ and $\overline{\...
2
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2answers
55 views

Riesz representation for products?

Given is a continuous linear functional $T:C_c^0(\mathbb{R})\otimes C_c^0(\mathbb{R}) \rightarrow \mathbb{R}$ where $C_c^0(\mathbb{R})$ is the space of continuos functions with compact support. Since $...