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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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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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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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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 ...
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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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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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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}...
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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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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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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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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. ...
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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 *-...
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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-...
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1answer
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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 ...
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1answer
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If $\operatorname{ran}A$ is closed, is then $\operatorname{ran}A^2$ closed?

Let $X$ be a Banach space or Hilbert space and $A : X\to X$ is bounded operator such that If $\operatorname{ran}A$ is closed. Is then also $\operatorname{ran}A^2$ closed? I think not. Can anyone ...
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The shift operator S's inverse operator [on hold]

If S is the shift operator on $l^2$ which means $Se_n=e_{n+1}$ and $|\lambda|>1$, then $$||(\lambda-S)^{-1}||=\frac{1}{|\lambda|-1}$$ I need to prove this statement. We can easily get that $||(\...
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Show that the semigroup given by the product of two semigroups is generated by the product of the generators

Let $X_n,Y$ be $\mathbb R$-Banach spaces for $n\in\mathbb N$, $A_n\subseteq X_n\times X_n$ and $B\subseteq Y\times Y$ be linear and dissipative with $\mathcal R(\lambda-A_n)=X_n$ and $\overline{\...
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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{\...
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Norm of an operator in different spaces

Given three Banach spaces $X$,$Y$ and $Z$ such that $Y \subset Z$ with dense and continuous embedding, we consider $T\in \mathcal{L}(X,Y)$. Is it true that $$ \|T \|_{\mathcal{L}(X,Y)} = \|T\|_{\...
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1answer
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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 $||ap||\le \sqrt {||ap||}$. ...
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Hille-Yosida theorem for multi-valued dissipative linear operators

Let $E$ be a $\mathbb R$-Banach space, $A$ be a multi-valued dissipative linear operator on $E$ and $$A_0:=\left\{(x,y)\in\overline A:y\in\overline{\mathcal D(A)}\right\}.$$ Assume $\overline{\...
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1answer
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Showing An Operator is Unitary

Let $T = T^*$ be operators and $\lVert T\rVert \leq 1$. Define an operator $U := T + i(I - T^2)^{1/2}$. Then U is a unitary operator. I know that I need to show that $U^*U = UU^* = I$. So I must find ...
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1answer
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If $A_0=\{(x,y)∈\overline A:y∈\overline{D(A)}\}$, then $R(λ-A_0)=\overline{D(A)}$ if and only if $R(λ-\overline A)⊇\overline{D(A)}$

Let $E$ be a $\mathbb R$-Banach space, $A$ be a multi-valued dissipative linear operator on $E$, $$A_0:=\left\{(x,y)\in\overline A:y\in\overline{\mathcal D(A)}\right\}$$ and $\lambda>0$. How can ...
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1answer
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How do we call a multi-valued operator $L$ with $∃c>0:∀y\in Y:∃x\in X:(x,y)\in L\text{ and }\left\|x\right\|_X\le c\left\|y\right\|_Y$?

Let $X,Y$ be $\mathbb R$-Banach spaces and $L$ be a multi-valued linear operator from $X$ to $Y$ with $$\exists c>0:\forall y\in Y:\exists x\in X:(x,y)\in L\text{ and }\left\|x\right\|_X\le c\left\|...
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1answer
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If $B$ is a closed subspace of a Banach space $E$ and $L_n,L$ are bounded linear operators on $E$ with $L_nB⊆B$ and $L_n\to L$, then $LB⊆B$

Let $E$ be a $\mathbb R$-Banach space $B$ be a closed subspace of $E$ $(L_n)_{n\in\mathbb N}\subseteq\mathfrak L(E)$ with $$L_nB\subseteq B\;\;\;\text{for all }n\in\mathbb N\tag1$$ and $L\in\mathfrak ...
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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 $...
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1answer
34 views

Invertibility of weighted shift operator.

A linear operator $T$ on a (complex) 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=...
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+50

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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1answer
27 views

Fundamental property of Green's function is violated

$$\langle x| D|x'\rangle=D_x\langle x|x'\rangle=D_x\,\delta(x-x')$$ $DD^{-1} = I$ , Where '$I$ ' represents the Identity position representation of this equation is $\langle x|D|x'\rangle \langle ...
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How boundary conditions for a green function is same a original function

I am trying to study green's function.suppose we have a differential equation, $D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
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Compactness of a special kind of Integral operators

Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{ & K:{L^2}(0,1) \to {L^2}(0,1) \cr & f: \to (Kf)(x) = \int\limits_0^1 {k(...
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1answer
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Extension of bounded linear operator in Hilbert Spaces

Let $H=(H,(\cdot,\cdot))$ be a Hilbert space and $G\subset H$ equiped with a norm of $H$. Let $F=(F,||\cdot||_{F})$ a Banach space and $S:G \longrightarrow F$ a bounded linear operador. Prove that, ...
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Rescaling of argument of kernel of a Fredholm determinant. How does this effect the det?

Suppose we have a trace class kernel $K_x(z,z')$ acting on $L^2((ax,bx))$ ( square integrable functions on the interval $(ax,bx)$, where $x>0$ is a positive parameter). In trying to study the ...
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1answer
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Show that the norm of this operator is equal to 1

Let $H$ be a Hilbert space and $P$ a projection $H \rightarrow H$ ( a bounded linear operator on $H$ such that $P^2=P$ and $P$ is not equal to $0$) I showed that $ ||P|| \ge 1$ and that $P$ is auto ...
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Conclude $X^n_t\to X_t$ weakly from knowing that $X^n_0\to X_0$ weakly and convergence of the corresponding generators

Let $E$ be a locally compact separable metric space $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $C_0(E)$ with generator $(\mathcal D(A),A)$ $(T_n(t))_{t\ge0}$ be a uniformly ...
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Multipliers and corners of $C^*$-algebras

Let $A$ and $B$ be $C^*$-algebras. Suppose that there exists a projection $p$ in $\mathcal{M}(B)$, the multiplier algebra of $B$, such that $A=pBp$. That is, $A$ is a corner of $B$. Question: Is it ...
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Iteration of multiple derivatives (infinitesimal generator)

Suppose we have a smooth and good enough function $f(x):R\rightarrow R$. Let us define an operator $H$ such that $$ H(g(x)) = \frac{dg(x)}{dx} f(x) + \frac{1}{2}\frac{d^2g(x)}{dx^2}C, $$ for ...
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When is a weak topology Baire?

Given a locally compact space $X$, and a set of functions $A\subset C_b(X)$ (where $C_b(X)$ is the set of continuous bounded complex functions)- when can we say that the weak topology generated by $A$ ...
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2answers
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Inequality related to the norm of operator

Let $T$ a bounded operator acting on a complex Hilbert space $\mathcal{H}$. It is well know that the operator norm of $T$ is given by: $$\|T\|=\sup\left\{\frac{\|Tx\|}{\|x\|}\,;\;x\in \mathcal{H},\,...
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1answer
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An example of a non unitary isometry on $L^2(\mathbb{R})$?

I would like to see one or preferrably two isometries on $L^2(\mathbb{R})$ which are non surjective (equivalently non unitaries)? Thanks in advance. Math.
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The second dual of a von Neumann algebra

Let $A$ be a von Neumann algebra and let $A^{**}$ be the second dual space of $A$. Is this true that $A^{**}=A$ ?
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Commutant of a weighted shift operator

I have this theorem I would like to prove in a different style from that in the paper by Allen. Shelds Let $T$ be a weighted shift operator with weight sequence $\{w_n\}$. It is known that $T$ is ...
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Green function derived from the idea of position representation in quantum mechanics

I am trying to study green's function by using a lecture by NPTEL its link is https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s I don't get this step by him at the minute of ten. $D_x G(x,x^{'}...
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On the connections between the spectral theorem and functional calculus for normal operators.

I am interested on some connections between the spectral theorem for normal operators and a corresponding functional calculus which can be assigned to them. I'll begin by recalling these two results, ...
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12 views

support projection of a sum of positive operators

Let $T$ and $S$ be positive selfdjoint bounded operators on a Hilbert space $H$. Let $s(T)$ be the support projection of $T$. Do we have $$ s(T+S) \geq s(T) ? $$ (and also $s(T+S) \geq s(S)$)
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1answer
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Position representation of an operator

$$\langle x| M|x'\rangle=M(x)\langle x|x'\rangle=M(x)\,\delta(x-x')$$ I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general operator $M $? $\...
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1answer
65 views

Proving that positive linear maps from or to C(X) are completely positive

I want to prove the following: Suppose A is a unital C*algebra and X is a compact Hausdorff space, then any positive map $\phi: A \to C(X) $ and any positive map $\psi: C(X) \to A$ is completely ...
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184 views

Calculate the operator norm of $A: L^2[0,1] \to L^2[0,1]$ defined by $(Af)(x):=i\int_0^x f(t)\,dt-\frac{i}{2} \int_0^1 f(t) \,dt$

I want to calculate the operator norm of the operator $A: L^2[0,1] \to L^2[0,1]$ which is defined by $$(Af)(x):=i\int\limits_0^x f(t)\,dt-\frac{i}{2} \int\limits_0^1 f(t)\, dt$$ I've already shown ...
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0answers
29 views

Injective Weighted Shft Operator

A linear operator $T$ on a (complex) 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=...
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0answers
40 views

If $E^\ast$ is the one-point compactification of $E$, can we identify a contraction semigroup on $C_0(E)$ with a contraction semigroup on $C(E^\ast)$?

Let $E$ be a locally compact Hausdorff space, $E^\ast=E\cup\left\{\infty\right\}$ denote the Alexandroff one-point compactification of $E$, $(T(t))_{t\ge0}$ be a strongly continuous contraction ...
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1answer
38 views

A weakly continuous semigroup of operators on a Banach Space is Strongly Continuous

The following is an exercise from Dunford & Schwartz (1958) page 653. Let $X$ be a Banach space and $T_t:X \to X$ be a semigroup of bounded operators indexes by $\mathbb{R}_{\geq0}$. Suppose it's ...