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# Questions tagged [operator-theory]

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

695 questions
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### Quasinilpotent vectors of Newton potential vanish

Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential $$T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy.$$ It is well know ...
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According to this answer: Take copies of Hilbert spaces: $H_1,H_2,H_3,...$ and $K_2,K_3,K_4,...$. Let $A_1$ be a partial isometry which is a shift $H_1 \rightarrow H_2 \rightarrow H_3 \rightarrow ...... 1answer 189 views ### Quasinilpotent , non-compact operators If$X$is a separable Banach space and$(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on$X$such that$||T^n||^{1/n}<\epsilon_n$for all$n$? I suspect the answer ... 0answers 66 views ### Relation between spectrum of an operator and its cut down Let$T$be a self-adjoint operator in$\mathcal{H}$with spectrum$\sigma(T)$, Let$P$be a projection in the commutant of$vN\{T\}$, the von Neumann algebra generated by$T$, question what is the ... 0answers 90 views ### examples of MF algebras [closed] Can anyone show me concrete examples of unital MF algebra and non-unital MF algebra respectively? Thanks! 0answers 70 views ### Measurability of the heat semigroup in$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)$... 0answers 33 views ### A question on the Non-degenerated bilinear form [migrated] Prove that$<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-... 1answer 64 views ### 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 ... 2answers 87 views ### 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-... 1answer 69 views ### 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 ... 0answers 70 views ### 1D Schrödinger Equation with Measure-Valued Coefficients I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on$[0,1]$with Hamiltonian given by the following: $$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}... 0answers 92 views ### Compact operator Let k:[0,1]^2 \to [0,1] be a measurable function. Define K:L^2([0,1])\to L^2([0,1]) to be the operator:$$ (Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}... 0answers 136 views ### 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 ... 0answers 116 views ### A special sequence I m looking for a sequence$(f_j)\in C^\infty(\Bbb{R})$such that $$\int^\infty_0\Big|\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\Big|^2rdr\to 0,$$ and$$\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1\... 2answers 122 views ### Non-isolated ground state of a Schrödinger operator Question. Does there exist a dimension$d \in \mathbb{N}$and a measurable function$V: \mathbb{R}^d \to [0,\infty)$such that the smallest spectral value$\lambda$of the Schr?dinger operator$-\...

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