All Questions
Tagged with perturbation-theory operator-theory
28
questions
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24
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Question related to isolated eigenvalue of a Hermitian operators
This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
- 78
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38
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Number of distinct discrete eigenvalues of a self adjoint operator after compact perturbation increases.
Let $T$ be a self adjoint operator on a complex separable Hilbert space $H$. Let $K$ be self adjoint compact operator. So, $T+K$ is also self adjoint operator. I want to know for what $K$ the number ...
- 78
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42
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Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator
Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator. In particular, is there any way we can say that the element of essential spectrum is an eigenvalue of ...
- 78
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1
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90
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Perturbation of a maximal dissipative operator by a non-negative self-adjoint operator.
Let $A$ be a maximal dissipative operator in a Hilbert space $\mathcal{H}$, and consider $B$ a self-adjoint operator such that
$$ \langle B\xi,\xi \rangle \geq 0\ , \quad \xi\in \mathcal{H}\ .$$
Does $...
- 87
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Reference request: Let $T$ be a self-adjoint operator, and assume that $V$ is compact and self-adjoint. Then $\sigma_{ess}(T + V ) = \sigma_{ess}(T)$.
I am looking for a proof of the following result.
Let $T$ be a self-adjoint operator, and assume that $V$ is compact and self-adjoint. Then $\sigma_{ess}(T + V ) = \sigma_{ess}(T)$.
Here the context ...
- 520
2
votes
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answers
89
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Alternative reference request for Kato's Perturbation Theory
I am studying Analytic Perturbation Theory from Kato's book.But sometimes I find it really difficult to follow.(e.g. In Chapter 2 , he directly talks about algebraic singularities without defining it ...
3
votes
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answers
359
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Perturbation of the spectrum
In a lecture note, in the demonstration of a result it is said that "using analytic perturbation theory" it is possible to deduce certain things. The reference is Kato's classic book, "...
- 2,797
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46
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Regular Perturbation theory for $L=L_0+\epsilon ix$, where $L_0$ is an unbounded self-adjoint operator, on $\mathbb R$
I am looking to understand the spectrum of the following operator:
$L_{\epsilon}[u](x)=L_0[u](x)+i\epsilon x u(x)$ on $\mathbb R$. Here $L_0$ is a negative semi-definite self-adjoint operator that has ...
2
votes
1
answer
509
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Eigenvalue perturbation of hermitian matrices
For $\rho$ and $A$ hermitian matrices, $t$ a scalar parameter, $f_i(\rho)$ denoting the $i-$th eigenvalue of the matrix $\rho$ and $|x_i\rangle$ the normalized eigenvector of $\rho$ corresponding to $...
- 181
1
vote
2
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197
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Computing the resolvent of a rank one projector
I'm reading through this paper in dynamical systems, that's using a bit of perturbation theory and operator theory. The authors make the following claim:
Now, since $P_0$ is a projection of rank one, ...
- 125
2
votes
1
answer
567
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Analytic eigenvalues
Is the following conclusion true?
Suppose $A,B$ are $n\times n$ complex Hermitian matrices. Then there exists real analytic functions $\lambda_i:\mathbb R\to \mathbb R$ where $1\leq i\leq n $ such ...
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votes
1
answer
60
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Smallest eigenvalue of sum of two unbounded operators
Suppose $T,S:D(\mathcal H)\to \mathcal H$ are two unbounded operators with discrete spectrum consisting eigenvalues $0<\lambda_1(T)\leq\lambda_2(T)\leq\dots$ and $0<\lambda_1(S)\leq\lambda_2(...
- 3,976
4
votes
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79
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$D((A+B)^*)= D(A^*)$ if $B$ is $A$-bounded with $A$-bound $0$
Let $A:D(A) \subseteq H \to H$ and $B:D(B) \subseteq H \to H$ be linear operators on a Hilbert space $H$ such that $A$ is a closed densely defined operator and $B$ is relatively bounded with respect ...
- 1,048
2
votes
1
answer
127
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Showing that $\lambda - (A + B)$ has dense range
Let $A$ be the generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$ of contractions on a Banach space $X$ and $B \in \mathcal L(X)$ a bounded operator. To apply some approximation formula I want to show ...
- 5,137
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Conditions for Boundedness of Spectral Measures of Perturbations of Self-Adjoint Operators?
Suppose $A$ is an unbounded self-adjoint operator in a Hilbert space $H$ with discrete spectrum $$\lambda_0 < \lambda_1 < \cdots$$ bounded below with lowest eigenvalue $\lambda_0$, lowest ...
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