# Tagged Questions

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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### If an operator has a cyclic vector, then its co-rank is at most $1$

Prove that if an operator on a Hilbert space has a cyclic vector, then its co-rank is at most $1$. My attempt: If an operator $T$ on Hilbert space $H$ has cyclic vector $u$, and $v \in$ $(TH)^\perp$,...
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### A matrix has a cyclic vector iff every eigenvector is of geometric multilplicity 1

In a finite dimensional space, prove that a matrix has a cyclic vector iff every eigenvalue is of geometric multilplicity 1. I can show that if a matrix has a cyclic vector, then every eigenvalue is ...
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Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain $D(A)... 0answers 18 views ### Reference for measures of commutativity needed I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if$A$and$B$are two$n \times n$Hermitian matrices, and$[A,B]=C$. I'd like a ... 1answer 90 views ### Why has the Stein operator for normal approximations the form$(\mathcal Af)(x)=f^\prime(x)-xf(x)$? My Question: Why has the Stein operator$\mathcal A$for normal approximations the form$(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ... 2answers 30 views ### A relation between the domain of$A$and the domain of$\bar A$Let$A$be an operator: $$A:D(A)\to R(A)$$ where$D(A)$and$R(A)$are respectively the domain and the range of$A$and they are subspaces of a Hilbert spcae$(H,\|\|)$. Suppose that$A$is a ... 2answers 75 views ### proving that$\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that$\text{rge}\,A\subset\text{aff}\,C$and for$\epsilon>0$claims that$\epsilon^{-1}(C-\text{rge}\,A)\...
Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...