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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Does every closed, densely operator in a Banach space have an closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. ...
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1answer
77 views

Wave Operators: Summary

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
40 views

Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
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56 views

Boundedness of linear operators in $L^p$

I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces: i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$. ii) There exists ...
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53 views

Hilbert- Schmidt class is an ideal

Definitions: 1 - An operator $y\in B(H)$ is said to be of trace class if $y$ is compact, and also $\sum|\alpha_n| <\infty$ where $\alpha_n \in \sigma(y)$ and $y$ has a representation $\sum ...
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1answer
80 views

weak convergence of $L^2$ implies weak convergence of $W_0^{1,2}$ (up to a subsequence)?

In the paper that I am reading, it says that if $\{u_n\}$ are bounded in $W_0^{1,2} (\Omega)$ (bounded $\Omega\subset \mathbb{R}^N$) and $u_n \rightharpoonup u$ weakly in $L^2 (\Omega)$, then there ...
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2answers
89 views

Compact operator space is the greatest ideal of $B(H)$

Suppose $H$ is a separable infinite dimensional Hilbert space. Show that if $A\in B(H)$ is noncompact, then there exist two operators $B,C$ such that $BAC=1$. Clearly if $A$ is invertible it holds, ...
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1answer
61 views

Differential operator a bounded operator or not?

Is the operator $T$ a bounded operator mapping $T: H^n([0,\pi]) \rightarrow H^{n-1}([0,\pi])$ ($H^n$ is the n-th Sobolev space with respect to $L^2$) or not? The operator itself is given by ...
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46 views

Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
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55 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
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108 views

Trace class operator

Let $A\in B(H)$ and $\sum_{E}|\langle A e,e\rangle|< \infty$ for every orthonormal basis $E$. Show that $A$ is a trace class (means $\sum_E \langle |A|e,e\rangle < \infty$). I can not prove it. ...
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33 views

Compact operator and a sot convergent sequence of operators

The following is an exercise of Conway's operator theory: I proved all parts of this exercise except $\|KT_n\| \to 0$. I can easily prove $\|KT_n^*\|\to 0$, but do not have any idea to prove ...
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1answer
153 views

Show that the operator $(x_n)_n\mapsto (\frac{x_n}{n}) $ is compact

I want to show that the following operator is compact: $$T:\mathbb \ell^p\rightarrow \mathbb \ell^p, \text{ }(x_n)_n\mapsto(\frac{x_n}{n})_n \text{ } 1\leq p<\infty$$ Its the first time that ...
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1answer
93 views

On Fredholm operator on Hilbert spaces

Let $u: H \to H'$ be a continuous linear operator and $H,H'$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite ...
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28 views

Irridicible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

Let $A$ be an irreducible C*-subalgebra of $B(H)$ and $p$ be a nonzero projection in $B(H)$. Suppose $pAp=\Bbb C p$, show that $p$ is rank one. I do not have any idea about it. Please give me a ...
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119 views

Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
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1answer
23 views

Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N $ and $N$ is finite, then $M$ is finite. I think it should say ...
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3answers
80 views

Why is $R-\lambda$ invertible for $|\lambda|<1$

I got the following question: Why is $R-\lambda$ invertible for $|\lambda|>1$ and not invertible for $|\lambda|\leq1$ ? R is the right shift operator on $\mathfrak{l^2}$
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2answers
64 views

Fock Space: NESS

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: ...
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2answers
93 views

From continuous to bounded Borel functions

I know that we can extend the functional calculus of bounded self-adjoint operators to bounded Borel functions. I want to do the same for unbounded self-adjouint operators. Therefore assume that $T$ ...
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48 views

Fock Space: Formal Adjoints

Problem Given a pre-Hilbert space $\mathcal{H}$. Consider unbounded operators: $$S,T:\mathcal{H}\to\mathcal{H}$$ Suppose they're formal adjoints: $$\langle ...
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54 views

Prove of some properties about unitary operators [closed]

Let $X$ be a hilbert space and $T\in L(X)$ be an unitary operator. Show (1) $\sigma(T)\subset\{\lambda \in \mathbb C:|\lambda|=1\}$ (2) for $\lambda \in \mathbb C$ with $|\lambda|\neq1$ holds: ...
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45 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
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157 views

Finite rank volterra operator

I am wondering when a Volterra integral operator $V_K:L_2(0,1)\to L_2(0,1)$ is a finite rank operator: $$V_Kf=\int_0^xK(x,y)f(y)dy$$ thanks in advance for your help
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1answer
145 views

Continuous operator between Banach spaces, closed range

I have some problems proving the following: $T: X \rightarrow Y$ is a continuous, linear operator between Banach spaces. Prove that $T$ is surjective $\iff$ $T^* : Y^* \rightarrow X^*$ is injective ...
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49 views

Nature of the infinite differential sum operator?

Consider the operator $$ Hf = f + f' + f'' +\cdots = \sum_{i=0}^\infty \left[ \frac{d^i f}{dx^i}\right] $$ I am trying to determine what $ Hf $ is entirely in terms of $f$. I note the following ...
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77 views

Invariant subspace of bounded self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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1answer
56 views

Invariant subspace of self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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30 views

Ask for the name of a condition on commutator of two operators

Let $T, S$ be two bounded linear operators on a Hilbert space. I wonder whether there is a standard way referring the following condition: $$ \text{The commutator $[T, S]$ is in the Hilbert-Schmidt ...
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61 views

Von Neumann algebra generated by a subalgebra

Let A be a C*-algebra of operators on a Hilbert space H. Show that if $A\subset K(H)$, then $\{A'\cap K(H)\}'\cap K(H) = A$ I do not have any idea about it. Please give me a hint. Thanks.
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Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
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1answer
38 views

Show that a nondegenerate *-Banach algebra is a C*-algebra

Takesaki in his operator theory says A C*-algebra $M$ of operators on Hilbert space $H$ means a nondegenerate ( $\text {cl} (MH) = H$) $*-$ subalgebra of $B(H)$ which is closed under the uniform ...
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99 views

Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
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Question about $C_0(X)$-algebras and $C_b(X)$.

Let $X$ be a locally compact Hausdorff space. Denote by $C_0(X)$ its C*-algebra of continuous functions that vanish on infinity and by $C_b(X)$ its C*-algebra of bounded functions. Now, let $A$ be a ...
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289 views

Every normal operator on a separable Hilbert space has a square root that commutes with it

Show that every normal operator on a separable Hilbert space has a square root that commutes with it. Uniqueness? My attempt: Let $T$ be a normal operator. By polar decomposition $T=U|T|$ where ...
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1answer
47 views

Image of a projection

Show that $\lambda$ is an eigenvalue for normal bounded linear operator $N$ on Hilbert space $H$ with spectral measure $E$ iff $E(\{\lambda\})\neq 0$, in which case the range of $E(\{\lambda\})$ is ...
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104 views

On the weak closure of a sequence of projections

Let $H$ be a Hilbert space with $\text{dim}=\infty$ , and $\{e_n\}$ be an orthogonal sequence of projections in $B(H)$. Show that $\{\sqrt{n}e_n ; n\geq 1\}$ does not admit a subsequence converging to ...
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1answer
43 views

Unique solution of $(L+\alpha I)z=y$

Let $L:X \rightarrow X$ be a bounded linear operator with bounded-inverse. How we can show that $(L+\alpha I) z= y $ has a unique solution for sufficiently small $\vert \alpha \vert$? If $Lx=y$, ...
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78 views

Power of positive operator

Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in ...
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2answers
77 views

Bounded or unbounded operator?

Consider the operator $A\colon\ell^2\to\ell^2$ defined by $Au=(ku_k)$. Normally, this is not well-defined, since $(1/k)_{k=1}^\infty\in\ell^2$ but $A(1/k)=(1,1,\ldots)\notin\ell^2$; however, if one ...
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149 views

K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
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Sot convergence of a net

The following are exercises of Conway's operator theory: I proved both exercises, but I confused about this point that in exercise 8, $T_i\to 0$ (sot), so based on exercise 6, $T_i^2 = T_i.T_i\to ...
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States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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Weak operator topology is the smallest topology on $B(H)$

Show that weak operator topology is the weakest locally convex topology on $B(H)$ such that every $\phi\in F(H)$ is continuous. (F(H) means finite rank operators on $H$). To show it , let $\tau$ ...
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Closed unit ball of $B(H)$ with wot topology is compact

The following is a Theorem of Conway's operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact ...
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83 views

An inequality for positive operators

Let $S$ and $T$ be positive operators on a Hilbert space $\mathcal{H}$. Suppose that $S \le T$. Since the square root function is operator monotone, it follows that $S^{1/2} \le T^{1/2}$. Does the ...
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67 views

Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...
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1answer
145 views

On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$

The following is a theorem of Takesaki's operator theory: In this proof, weak topology means weak operator topology. I'm wonder why the theorem holds just for bounded parts of $B(H)$ and also ...
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1answer
174 views

Using lemma in proof

Hi please view the attachment. I am interested in how Lemma 1.11 is used in the proof of Theorem 2.10. Based on the statement of Lemma 1.11 it seems that in order to use Lemma 1.11 in we require ...
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59 views

Operator in Denominator

So I chanced upon this statement, and I'm not sure what is happening: $$ \left(c-\frac{1}{b}\frac{\partial}{\partial ...