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.

6,372 questions
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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 ...
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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 ...
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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?
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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 ...
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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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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 ...
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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 ...
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 $T(x)= \sum\langle x,... 2answers 18 views How can I show that this set is a closed subspace Let$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$as the kernel of a continuous linear ... 1answer 20 views Green function, determine$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: $$-(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}... 1answer 787 views Frechet/Gateaux differentiability of an integral operator L^2 \rightarrow R Let f: R \rightarrow R be a continuously differentiable function on the real numbers (if needed also infinitely many often differentiable). Define the Operator F : L^2([0,1]) \rightarrow R for x ... 0answers 24 views Representing linear operator with matrix I've came across the following question in a book that I read about Hilbert spaces and linear operators - Where exercise 1 is - So the book has answers, and for 13a it is - what I don't ... 0answers 17 views Equivalent definition for approximate point spectrum Theorem Let A be an operator on a Hilbert space H. The approximate spectrum denoted by \sqcap (A). The following are equivalent: 1) \lambda\in \sqcap (A). 2) There exists a sequence \{f_n\}... 0answers 44 views Book recommendation request on Spectral theory Can someone please recommend to me a text that deals on spectral theory from the scratch covering the parts of a spectrum (approximate, point and compression) explicitly. Theorems and properties. ... 0answers 26 views Show an inclusion for the range of a multi-valued dissipative operator Let Z be a \mathbb R-Banach space and C be a multi-valued dissipative linear operator on Z, i.e. C is a subspace of Z\times Z with$$\forall\lambda>0:\forall(z,z')\in C:\left\|\lambda z-... 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 *-... 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 ... 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$||ap||\le \sqrt {||ap||}$. ... 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{\...
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 \$...