# Questions tagged [operator-theory]

Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

696 questions
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### examples of MF algebras [on hold]

Can anyone show me concrete examples of unital MF algebra and non-unital MF algebra respectively? Thanks!
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### Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$. It is known that $S(t)$ ...
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### A question on the Non-degenerated bilinear form [migrated]

Prove that $<C(G), C(G)>$ is dual system with bilinear form $$\int_G \phi(x)\psi(x)dx$$ and $\psi(x),\psi(x)\in C(G)$ so here i was trying bilinear form im not getting how to prove non-...
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### Existence of a bounded operator which satisfies the discrete product rule

Is there a bounded self-adjoint operator $H$ acting on $\ell^2(\mathbb{Z})$ such that for all sequences $u,v\in \ell^2(\mathbb{Z})$ $$H(uv)=(Hu)v+u(Hv)$$ where uv is the pointwise product. This is ...
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### Polar decomposition of tensor product of operators in von Neumann algebra

If $T=V|T|\text { and } S=W|S|$ is the polar decomposition of $T$. Is it true that the polar decomposition of $T\otimes S$ is $T\otimes S=(V\otimes W)(|T| \otimes |S|)$. If $T$ and $S$ are self-...
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### power of an infinitesimal generator of a $C_0$ semigroup in a Banach spaces

Let $A$ be the infinitesimal generator of a $C_0$ semigroup of linear operators in a Banach space. Let $n$ pe a positive integer, $n\geq2$. Is $A^n$ closed? Here (setting $A^1$ $:=$ $A$, and ...
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### Blocksum induces a unital H-space structure on the space of Fredholm operators

Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
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### What is the story behind this Hilbert space in the definition of Hilbert Modules

Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück: A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...
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### A single vertex operator - bare bones explanation?

There is an ocean of literature (and a sea of popular texts inside) on vertex algebras, including quite a lot of Q & A here on MO, and I am trying to read some random selections from time to time. ...
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### Kernel of compact operators [closed]

Since a compact operator $K$ on an infinite dimensional separable Hilbert space $H$ can't be invertible, the spectrum of $K$ must contain zero $0\in \mbox{sp}(K)$. However, $K$ could be injective (e.g....
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### Stone–von Neumann theorem?

The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR) $$U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t$$ ...
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### Interpolation of a trilinear functional

Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
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### Gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded. To be more precise, a ...
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### Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
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### Are these kinds of “crossed product” studied?

Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...
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### About crossed product of the group von Neumann algebra

Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...
### Convergence of self-adjoint elements in $\sigma$-weak topology
In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?