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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Completely continuous map is not homotopy with antipodal map

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Now, $E$ is a infinity dimensional ...
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4 views

Understanding certain symbols in “Non-Positive Partial Transpose Sub spaces Can be as large as any Entangled Subspace”

This is a link to a paper entitled "Non-Positive Partial Transpose Sub spaces Can be as large as any Entangled Subspace" by Nathaniel Johnston. I have issues understanding the notation used in this ...
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1answer
35 views

Does an essentially self-adjoint operator have the same kernel as its closure?

Let $H$ be a Hilbert space and let $A : D(A) \subset H \to H$ be an essentially self-adjoint operator. Let $\overline A$ be the unique self-adjoint extension of $A$. Question: Is it true that ...
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32 views

definition of block diagonal operator on a hilbert space

I 'm stuck with the definition of block diagonal operators on hilbert spaces. Def.: A bounded linear operator $T$ on a hilbert space $H$ is called block diagonal if there exists an increasing ...
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53 views

What is the $C^*$-algebra generated by a normal operator?

The following is the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I don't find the definition for the $C^*$-algebra generated by a normal operator in the book. ...
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19 views

Norms under Conjugation by Projection Opertaros

I was reading about equivalent forms of the Kadison Singer Problem, and while looking at the Feichtinger Conjecture, I came across the claim that, for a projection operator $P$ and a self-adjoint ...
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21 views

Does pointwise nilpotency imply global nilpotency?

Is there a compact Haussdorf space $X$ and $C^{*}$ algebra $A$ with a continuous map $f:X\to A$, such that $f(x)\in A$ is a nilpotent element, $\forall x \in X$, but $f$ is not a ...
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24 views

Equivalence of Definitions of Strong Operator Topology

I have a couple questions about how we define the strong operator topology on $\mathscr{B} (H)$ that I'm hoping someone can help me with. First, I thought that the strong operator topology was the ...
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75 views

Understanding bounded linear operators

The definition of a bounded linear operator is a linear transformation $T$ between two normed vectors spaces $X$ and $Y$ such that the ratio of the norm of $T(v)$ to that of $v$ is bounded by the same ...
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58 views

Does elementwise matrix inequality extend to norms?

The elements of $A$ and $B$ are non-negative and $A_{ij} \leq B_{ij} \; \forall \; i,j$. Is it true that $\Vert A \Vert_p \leq \Vert B \Vert_p$ ? The norm is the operator norm induced by the usual ...
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26 views

Show that the operator is NOT symmetric.

Show that the Sturm-Liouville operator $L$ in $L^2([a,b])$ given by $$L=\frac{1}{r(x)}\left(DpD+q\right)$$ is not symmetric. I'm assuming $p=p(x)>0$ and $q=q(x)\geq 0$, as described by the problem ...
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35 views

Show that $L$ is formally self-adjoint.

Consider the differential operator $$L=e^xD^2+e^xD,\;\;D=\frac{d}{dx},\;0\leq x\leq1,$$ $$u^\prime(0)=0,\;\;\; u(1)=0.$$ Show that $L$ is formally self-adjoint. I just don't really know how to start ...
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30 views

self-adjoint operators and linear dependence

Let $L$ be a self-adjoint differential operator given by $L=\frac{d}{dx}\left(a_2\frac{d}{dx}\right)+a_0$. If $u_1$ and $u_2$ are two solutions of $Lu=0$ and $J(u_1,u_2)=0$ for some $x$ for which ...
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24 views

concomitant and self-adjoint operator

If $Lu = u^{\prime\prime}+\omega^2u$, show that $L$ is formally self-adjoint and the concomitant is $J(u,v)=vu^\prime-uv^\prime$. Moreover, if $u$ is a solution of $Lu=0$ and $v$ is a solution of ...
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64 views

Projection operator in Banach space is continuous

Let $(X,||\cdot ||)$ be a Banach space with a Schauder basis, i.e. there exist $e_j \in X$, $j\in \mathbb{N}$, s.t. $||e_j||=1$ for all $j$ and every $x\in X$ can be uniquely represented as ...
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35 views

Operator/Matrix inequality

Let $A,B$ be non-negative matrices, such that $0\leq B\leq 1$. Is it true that $BAB\leq A$? (meant in the quadratic form sense) $A,B$ do not need to commute in general.
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37 views

Dual space ($X^{*}$) and $X^{**}$

According to my lecture notes (we're using Folland' Real Analysis textbook), if $X$ is a normed vector space, then $L(X,Y) = \left\lbrace \text{all bounded linear operators T} : X \rightarrow Y ...
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68 views

Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?

Let $X, Y$ be Banach spaces, $S \in B(Y^{*}, X^{*})$. Does such operator $T \in B(X, Y)$ exist so that $T^{*}=S$? I suppose that the answer should be - no. Are there any hints that might help in ...
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42 views

Proof compactness of adjoint operators

I am trying to understand a proof of the following statement: Given a complex B-space X and a compact operator $T:X\rightarrow X$, the adjoint operator $T^\ast:X^\ast \rightarrow X^\ast$ is compact as ...
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23 views

If $A$ is the Laplacian on $H^2(0,1)∩H_0^1(D)$, then the fractional power space $\mathfrak D(A^{r/2})=H_0^r(D)$ for all $r\in\mathbb R$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
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27 views

If $G$ is the Green's function of the Laplacian $A$ and $L$ is the integral operator with kernel $G$, then $L$ is the inverse of $A$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
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25 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
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49 views

What is the proof that linear operators can be treated as variables?

I understand what a linear operator is, but I don't understand why you can just treat it as a variable.
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29 views

Bilinear Form in Definition of Adjoint Operator

I'm working through some material in Analysis Now by Pedersen, and I'm a little confused about how the adjoint to an operator is defined. Pederson defines it by: If $X$ and $Y$ are normed spaces ...
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41 views

Example of a Projection Operator in $\mathbb{R^3}$

I'm looking for an operator $\hat P$ in $\mathbb{R^3}$ such that $\hat P^2=\hat P$ that is also Hermitian
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1answer
22 views

Is a bounded, linear, nonnegative and symmetric operator with finite trace on a Hilbert space Hilbert-Schmidt?

Let $U=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
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56 views

Approximate unit for a certain C*-algebra

Let $A$ be a C*-algebra and $p$ a projection in $A^{**}$. To prove $p$ is the smallest unit for $B: = \{a\in A; pap=a\}$, suppose $\{u_i\}$ is an approximate unit for $B$. It's easy to see $q: = ...
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49 views

Trace of multiplication operator on $L^2(\mathbb{T})$

Let $H=L^2(\mathbb{T})$, where $\mathbb{T}$ is the Torus. Consider a multiplication operator with a sufficiently nice function $f$. Is there somehow a formula like $$\mathrm{tr} M_f = C ...
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1answer
23 views

Prove Linear Operator $(T_{w}f)(x) = |\dot{w}|^{\frac{1}{2}} f(w(x)) $ Is Unitary

Let the inner product be given by: $$\langle f,g \rangle = \int_{-\infty}^{\infty} f(x)g^{\ast}(x)dx$$ Then I want to show: $\langle T_{w}f, T_{w}g \rangle = \langle f,g \rangle$ So we have: ...
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32 views

what is the operator name that for positive value returns the same and for negatvie returns zero?

there is an operator that currently I see in mechanical engineering topics that for positive values returns the same quantity inside, and for negative values, returns zero , it's like <> but a bit ...
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1answer
35 views

Operator theory problem

Show that operator $ T : C([0,1]) \to C([0,1]) $, where $$ Ty(t) = \int_{0}^{1} |x-t|^{-1/2}y(x) \ dx. $$ Calculate the norm T. Thank for help.
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50 views

Show that $S_n \to S $(weakly) and $T_n \to T$ strongly implies $S_nT_n \to ST$ weakly

Let $X,Y,Z$ be Banach Spaces. Let $T_n,T \subset BL(X,Y), S_n,S \in BL(Y,Z)$. Show that a) $S_n \to S $(weakly) and $T_n \to T$ (strongly) implies $S_nT_n \to ST$ (weakly) b) $S_n \to S ...
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38 views

Is the Hankel matrix defined by $\ln(k)/k$ a bounded operator on $\ell^2$?

I call a sequence $(b_k)$ of positive real numbers a Hilbert kernel if there is a constant $C > 0$ such that $$ \sum_{k=1}^\infty\sum_{j=1}^\infty b_{k+j}c_kc_j\,\le\,C\sum_{k=1}^\infty c_k^2 $$ ...
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48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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32 views

Functional analysis, linear operator problem

a) Let $D$ be closed subspace of $l^1$ defined as follows $$ D : = \left\{ x=(s_n)_{n \in \mathbb{N}} \in l^1 : \sum_{n=1}^\infty \left\vert {s_n \over n} \right\vert^2 < \infty \right\} $$ Is ...
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1answer
27 views

Proving an identity of the Lebesgue integral of a sublinear operator

Let $T$ be a sublinear operator. Suppose that $f\in L^{1}(\mathbb{R}^{n})$ and $A$ is a set of finite Lebesgue measure. Then I want to prove that for all $0<p<1$, we have ...
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1answer
63 views

The strong topology on $U(\mathcal H)$ is metrisable

The strong operator topology on a Banach space $X$ is usually defined via semi-norms: For any $x \in X$, $|\cdot|_x: B(X) \to \mathbb R, A \mapsto \|A(x)\|$ is a semi-norm, the strong topology is the ...
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The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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Is the range of a self-adjoint operator stable by its exponential?

Let $H$ be an Hilbert space, and $A \in L(H)$ be a bounded linear self-adjoint operator on $A$. We assume that $R(A)$, the range of $A$, is not closed. Is it true or not that $R(A)$ is stable by ...
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1answer
50 views

If $Q$ is an operator on a Hilbert space with $Qe_n=λ_ne_n$ for all $n$, then $Q^{-\frac 12}e_n=\frac 1{\sqrt{λ_n}}e_n$ for all $n$ with $λ_n>0$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $\mathfrak L(U)$ be the set of bounded and linear operators on $U$ $Q\in\mathfrak L(U)$ be nonnegative and symmetric ...
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39 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
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Solvable Schrödinger operator

I am currently looking for an example of a 1-dimensional Schrödinger Operator $-\frac{d^2}{dx^2} + V(x)$ with the following properties: 1) V(x) must be integrable 2) $-\frac{d^2}{dx^2} +V(x)$ must ...
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23 views

On connection of distance to finite rank operators and singular values.

Im trying to understand why the following; $s_n(T) = \inf\big\{\, \|T-L\| : L\text{ is an operator of finite rank }<n \,\big\}$ where $s_{n}$ are nth singular values, is a plausible claim to ...
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37 views

What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where ...
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28 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
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1answer
117 views

Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
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21 views

Image density in spectral theory

The operator $T$ is $\dfrac{d}{dt}$ and $$\left\{\begin{array}{lc}x'(t)-\lambda x(t)=-y(t)\\ x(0)=0\end{array}\right.$$ and the domain of $T$ is $D(T)=\{x\in L^2(0,\infty):\; x\; \text{absolutely ...
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32 views

Given a linear Hilbert-Schmidt embedding $ι$ between Hilbert spaces, prove that $ιι^*$ is a bounded, linear operator with finite trace

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $U_0:=Q^{\frac 12}(U)$, $$\langle u,v\rangle_0:=\langle ...
3
votes
1answer
37 views

Showing that a linear operator is closed

Consider the linear operator $A:D(A)\subset X\to X$. I want to show that $(\lambda I-A)$ is closed given that $(\lambda I-A)$ is invertible. We know that $(\lambda I-A)^{-1}$ is closed. Now if we let ...
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1answer
62 views

$T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$. [closed]

Let $T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$. Show that there is a constant $C$ such that ...