All Questions
10 questions
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
2
votes
0
answers
143
views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
0
votes
0
answers
109
views
The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
1
vote
0
answers
122
views
eigenvalues of integral operator with centered kernel
Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that
there is a reproducing kernel Hilbert Space $\...
- 519
3
votes
0
answers
201
views
Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions
Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
- 6,853
1
vote
0
answers
139
views
Property $(\mathcal{L}(\phi),\phi)\geq 0$ about a operator $\mathcal{L}$
Consider the operator $\mathcal{L} : H^2(\mathbb{T}_L) \subset L^2(\mathbb{T}_L) \longrightarrow L^2(\mathbb{T}_L)$ given by
$$\mathcal{L} = -\omega \partial_x^2+3\varphi^2-1,$$
that is
$$\mathcal{L}(...
- 205
1
vote
0
answers
181
views
Eigenvalues of product of operators
Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ ...
- 519
3
votes
1
answer
646
views
Eigenfunctions and eigenvalues of an operator defined by a certain integral
Let $k :[0,1]^2 \rightarrow \bf{R} $ be a kernel function definded by
$ k(x,y)= (1- max(x,y))^2 .$ Now, let $ L $ be a linear operator defined on $ L^2 [0,1] $ by $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
- 245
2
votes
0
answers
152
views
Relationship between eigenvalues of compact operators $A$ and $(A+A^*)/2$
A result from 'Topics in Matrix Analysis' by Horn & Johnson
(3.3.33) is the following: For $A\in \mathbb{M}_n$, $\sum_{i=1}^k Re \lambda_i(A) \leq \sum_{i=1}^k Re \lambda_i \big(\frac{A+A^*}{2}\...
- 21
6
votes
1
answer
1k
views
Is the sum of spectral projections a projection?
Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
$$P_{\{\lambda_1,...\lambda_n\}}=\frac{...
- 241