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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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Proving that $T$ is self-adjoint (or that $A,B$ are vertical to each other)

I'm facing the following problem: Let $X$ be a Hilbert space, $A,B$ be subspaces of $X$ such that $X=A\oplus B$ and $T\in B(X)$ such that $T\vert_A=\text{Id}_A$, $T\vert_B=-\text{Id}_B$. Prove that $...
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7 views

compute the multiplier algebra

If $A$ is a $C^*$ algebra, $\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is the essential ideal of $A$ ,then we have $\oplus_{c_0} M_{k(n)} (\mathbb{C} ) \subset A \subset \prod M_{k(n)} (\mathbb{C})$. ...
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9 views

construct an element in $\prod M_{k(n)} (\mathbb{C})$

Suppose $A$ is a $C^*$ algebra,$\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is a essential ideal of $A$ and there is an element $(x_n) \in A$ such that $(x_n) \in \prod M_{k(n)} (\mathbb{C})$ and $tr(x_n) \...
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0answers
15 views

Minimal or Maximal Von Neumann algebra contained in a given $C^*$ algebra

Let $A,B \subset B(H)$ be two concrete von Neumann algebra. Is $A\cap B$ a von Neumann algebra, too? What about the intrinsic analogy of this question, as follows: Let $C$ be a $C^*$ ...
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18 views

Finding matrix Operators

Give the matrix: ...
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1answer
18 views

comparison of multiplier algebras

Suppose $I$ is an essential ideal of a nonunital $C^*$ algebra $A$, can we compare $M(I)$ and $M(A)$,is $M(A)\subset M(I)$,where $M()$ denotes the multiplier algebra.
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49 views

Simple proof that an operator is compact

Let $\phi$ be a compactly supported smooth function on $\mathbb{R}$. I'm looking for a simple proof that the operator $$\left(-\frac{d^2}{dx^2}+x^2\right)^{-1}\phi$$ (where $x$ denotes multiplication ...
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1answer
33 views

construct a sequence of operators [on hold]

Let $(H_n)$ be a sequence of different finite dimensional complex Hilbert spaces, $A_n \in B(H_n),tr(A_n) \to 0(n \to \infty)$,but the norm of $A_n$ does not converge to 0,where $tr()$ is the standard ...
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14 views

strictly positive element in a separable $C^* algebra

If $A$ is a separable $C^*$ algebra,does there exist a strictly positive element in $A$?
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1answer
21 views

Is the set $\overline{\langle \{ f^{*}_{1}f_2: f_1,f_2\in J\}\rangle } $ equal to $B(\ell^2)$?

If $J=\left\lbrace f\in B(\ell^2): f^*(e_1)=0 \right\} $, is the set $\overline{\langle \{ f^{*}_{1}f_2: f_1,f_2\in J\}\rangle } $ equal to $B(\ell^2)$? ( $\ell ^2 $ is the Hilbert space $(\ell^2, \...
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36 views

a norm on a complex matrix

Suppose $A=\prod_n M_{k(n)}(\mathbb{C})$,where $M_{k_n}(\mathbb{C})$ is the space of all $k(n) \times k_n$ complex matrices,does there exist a norm $\| \|_0$ on $A$ (which is different from the ...
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24 views

What's exactly meant if one is saying that the closure of a multivalued operator is the generator of a $C^0$-semigroup?

Let $E$ be $\mathbb R$-Banach space $A$ be a subspace of $E\times E$ and $$\mathcal D(A):=\left\{x\in E\mid\exists y\in E:(x,y)\in A\right\}$$ $(T(t))_{t\ge0}$ be a contractive $C^0$-semigroup on $\...
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0answers
8 views

The set of 3-tuples numerical range of Hermitian operators is not convex

I am trying to find a counter example for the following claim. If $A_1, A_2, \dots, A_n$ are Hermitian operators then the set of all $n$-tuples of the form $\left \{ \left( \left < A_1 f, f \...
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2answers
102 views

Value of operator norm when $\mathcal{T}f(x)=\int^{x}_{0} f(t)dt$

Let $\mathcal{C}$ be the space of continuous functions on $[0,1]$ equipped with the norm $\|f\|=\int^{1}_{0}|f(t)|dt$. Define a linear map $\mathcal{T}:\mathcal{C}\rightarrow \mathcal{C}$ by $$ \...
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0answers
30 views

Hypercyclic operators in $L_p (0,\infty)$

I'm pretty new to this area of study so if there are logical lacune in my proof (I'm sure there are many) please let me know. This is material I'm self studying. I'm trying to adapt the methods used ...
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0answers
23 views

Equivalence of maximum in thenunit circle and null function on the basis

Given $\{e_n\}$ the unit basis of $c_0$ show that for $f \in c_0 ^*$ its equivalent: The function $|f|$ have a maximum in the unit sphere of $c_0$ Exists $m$ in $\mathbb{N}$ such that $f(e_n) = 0$ ...
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2answers
35 views

Linear operator in Lp and its norm [on hold]

Given $$g \in L_\infty[0,1]$$ show that, for 1 ≤ p ≤ ∞, $$ f \rightarrow f\cdot g$$ is a linear continuous operator of $$L_p[0,1] \to L_p [0,1]$$ and compute its norm
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1answer
18 views

pairwise orthogonal projections in an inseparable $C^* $ algebra

If $A$ is a separable $C^*$ algebra,then there are at most countable pairwise orthogonal projections.If $A$ is inseparable,how many pairwise orthogonal projections in $A$? If it has, is it ...
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15 views

$C_0$ direct sum of complex matrices

How many elements in the $C_0$ direct sum of complex matrices,say$\oplus_n M_n(\mathbb{C})$? $\oplus_n M_n(\mathbb{C})=\{(x_n)\in \prod_n M_n(\mathbb{C}):\|x_n\| \to 0\}$ I think there are ...
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1answer
32 views

Compact operators on $\ell^1$

Let $T\in \ell^1$, $Tx = (\lambda_1x_1,\dots,\lambda_nx_n,\dots)$. Want to show that if $T$ is compact, then $\lambda_n\to0$. I know for $p\in(1,\infty]$, canonical basis $e_n \rightharpoonup 0$ (so ...
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1answer
25 views

spectral properties of the shift operator

Let $E = \ell^2$, and consider the multiplication operator $M$: $$Mx = (\alpha_1x_1,\dots,\alpha_nx_n,\dots), \forall x\in E$$ where $(\alpha_n)$ is a bounded sequence in $\mathbb{R}$. And consider ...
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3answers
29 views

orthogonal projections in $C^*$ algebra

Suppose $A$ is an arbitrary $C^*$ algebra,can $A$ be linear spanned by all orthogonal projections of in it ? If not,is there a relationship between a $C^*$ algebra and all orthogonal projections in ...
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1answer
19 views

$T$ has not a closed range

Let $$T = {\rm diag}(0, 1, 0, \frac{1}{2!}, 0, \frac{1}{3!}, \dots)$$ Clearly $T$ is a positive operator on the Hilbert space $\ell_{\mathbb{N}^*}^2(\mathbb{C})$. I want to prove that $T$ has not ...
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1answer
56 views

How to show non existence of an operator $A: V \to V^*$

Let $V$ be a Banach space , $p \in (1,2)$ and $\mu >0.$ How can I prove that there does not exist an operator $A:V \to V^*$ such that $$\langle Au-Av, u-v\rangle \geq \mu \Vert u-v \Vert ^p,\qquad ...
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1answer
22 views

Operator Norm Question

Suppose I am interested in operators $T:X\to Y$, with $X$ and $Y$ both separable Hilbert spaces. The operator norm of such $T$ can then be taken as $$ \|T\| = \sup_{\|x\|_X\leq 1}\|Tx\|_Y. $$ Since ...
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0answers
29 views

compact self-adjoint operator [on hold]

If $T$ is a compact self-adjoint operator on a hilbert space $H$.Can $H$ be composed as follows: $H$ is the direct sum of all eigenspaces.That is to say, $H=\oplus H_i$,where $H_i=ker(T–\lambda_i ...
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1answer
12 views

Proving that $|x^*(x)| \leq \rho(x,Y)$

Exercise : Let $(X,\|\cdot\|)$ be a normed space, $Y$ a subspace of $X$ and $x^* \in X$ with $\|x^*\| \leq 1$ such that $x^*|_Y = 0$. Show that $\forall x \in X \setminus Y$, it is : $|x^*(x)| \leq ...
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1answer
56 views

finite rank projection

In Murphy's book,there is a statement:If p is a finite rank projection on $H$,then $pB(H)p$ is finite dimensional. My question:Given any $S\subset B(H)$,$S$ does not contain $Id_H$.Does there exist ...
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2answers
24 views

Measurable Operators

Suppose I have an operator valued function, $\omega\mapsto A(\omega)$; for each $\omega$, $A(\omega):X\to Y$, is a bounded linear operator with $X$ and $Y$ real Hilbert separable Hilbert spaces. ...
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1answer
15 views

a question on gns construction

If $\pi$ is a zero representation of $C^*$ algebra $A$,there is no state$\tau$ on $A$ such that $\tau(a)=(\pi(a)\xi,\xi)$. When we talk about GNS constuction,Should the zero representation be ...
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1answer
23 views

There exists a unique extension $\hat{T}$ of a bounded linear operator $T$.

I am trying to prove the following theorem : Theorem : Let $X,Z$ be Banach (normed) spaces and $Y$ be a dense subspace of $X$. Let $T:Y \to Z$ be a bounded linear operator. Then, there exists a ...
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1answer
20 views

When does an operator act diagonalisably on $V_{\lambda}$

Suppose I have an operator $H$ on a complex vector space $V$. Let $V_{\lambda}$ denote the generalised eigenspace (with respect to $H$) of a maximal eigenvalue $\lambda$ (maximal of real value). (...
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1answer
12 views

Is the set of closed range operators open?

As we know the set of surjective operators is open. Let $T\in B(\mathcal{H})$ be a closed range operator. Is there any $\delta>0$ such that for every $T'\in B(T, \epsilon) $, we have $T'$ is ...
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0answers
20 views

About Riesz theorem in two dimensions

If we have a polynomial $\sum a_n e^{inx}$, its analytic projection is $\sum_{n>0} a_n e^{inx}$. M. Riesz theorem tells us that the operator T that sends polynomials into its analytic projections, ...
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1answer
13 views

Characterization of positive operators below an operator

Let $T\in B(H)$ be positive. Is there a way to characterize all positive $S$ such that $S\leq T$?
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0answers
26 views

Range-Kernel Uncomplementation of Fredholm Operators

I need to find a normed space $X$ and a compact operator $K:X\to X$ such that $$X\neq N(I-K)\oplus R(I-K).$$ However, all my examples failed so far.
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1answer
30 views

$C([0,T],(L(X),\mathcal T_{\text{strong}})) = C([0,T],(L(X),\mathcal T_{\text{uniform}}))$?

Given a Banach space $X$, we have two topologies on the space of all bounded linear operators $L(X)$, one is uniform operator topology $\mathcal T_{\text{strong}}$, the other is strong operator ...
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1answer
15 views

nondegenerate representation of a $C^*$ algebra

Every representation $(\pi,H)$ of a $C^*$ algebra $A$ can be reduced to the case of a non-degenerate representation.Usually,we take $K=[\pi(A) H]$,then we get a non-degenerate representation $(\pi_K,...
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1answer
33 views

Finding the norm of the operator $M_g : L_p \to L_p$ where $g \in L_{\infty}$

Suppose $(X,\,\cal{A},\,\mu)$ is a measure space where $\mu$ is a finite measure, and $p \in (1,\,\infty)$. Say that we take some $g \in L_{\infty}$, and we define $M_g : L_p \to L_p$ by $$M_g (f) = ...
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1answer
24 views

Kernel and Range of an Integral Operator

I want to find the kernel and range of: $$T:C([0,1])\rightarrow C([0,1])$$ $$Tf(x)=\int_a^b(x-t)f(t)dt$$ I know I'm looking for $f(x)$ that satisfy: $$x\int_a^bf(t)dt-\int_a^btf(t)dt=0$$ but I'm ...
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1answer
32 views

$(\int_0^\infty T(t) dt) f(x) = (\int_0^\infty T(t)f dt)(x) = \int_0^\infty T(t)f(x) dt$?

Given a sequence of bounded operators $\{T(t)\}_{t\ge0}$ defined on the Banach space $C_0(\mathbb R^d)$ equipped with the supremum norm $|f|_0:=\sup_{x\in\mathbb R^d}|f(x)|$. Suppose $$\int_0^\infty \|...
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2answers
26 views

If $A+B\ge C$, can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?

Let $A$ and $B$ be two positive operators on a Hilbert space. $C$ is a positive operator with $A+B\ge C$. Can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?
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2answers
30 views

Show that the semigroup corresponding to the Yosida approximation is contractive

Let $(\mathcal D(A),A)$ be a closed densely-defined linear operator on a $\mathbb R$-Banach space $E$ such that $(0,\infty)$ is contained in the resolvent set $\rho(A)$ of $(\mathcal D(A),A)$ and $$\...
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1answer
28 views

operator norm of a matrix and trace of a matrix

Suppose $A=(a_{ij})$ is an $n\times n$ complex matrix, $\operatorname{tr}(A)=\displaystyle \frac{1}{n}\sum_{i}a_{ii}$. I wonder whether there exists a relationship between $\|A\|$ (the operator norm) ...
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0answers
20 views

How can we show that a Feller semigroup is strongly continuous?

Let $E$ be a normed $\mathbb R$-vector space $C_0(E)$ denote the space of continuous functions vanishing at infinity $(T(t))_{t\ge0}$ be a contractive nonnegative semigroup on $C_0(E)$ with $$T_tf(x)\...
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0answers
50 views

Asymptotic expansion of heat operator and Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$. It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
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2answers
30 views

Applying the uniform boundedness principle

I try to solve the following: Let $X,Y$ be Banach spaces and $B:X\times Y\to \mathbb{C}$ a linear map such that $$x_n\to 0\implies B(x_n,y)\to 0\ \forall y\in Y$$ $$y_n\to 0\implies B(x,y_n)\to 0\...
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1answer
25 views

An example of the Pseudo-inverse of an operator

Let $E$ an infinite dimensional complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$. Definition: Let $T \in \mathcal{L}(E)$. The Moore-Penrose inverse of ...
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0answers
21 views

Elliptic boundary condition eigenvalue problem

In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem Theorem 1....
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1answer
21 views

Proof of the integral representation of the resolvent

Let $E$ be a $\mathbb R$-Banach space $(T(t))_{t\ge0}$ be a $C^0$-semigroup on $E$ $(\mathcal D(A),A)$ denote the infinitesimal generator of $(T(t))_{t\ge0}$ $A_\lambda:=\lambda\operatorname{id}_{\...