All Questions
Tagged with functional-calculus operator-theory
11 questions
8
votes
2
answers
626
views
An inverse to functional calculus
Given a Borel function $f:\mathbb{R}\rightarrow\mathbb{R}\cup\{\infty\}$, functional calculus allows to calculate $F(x)$ for any unbounded selfadjoint operator $x$ on a Hilbert space $\mathcal{H}$, ...
- 176
5
votes
0
answers
227
views
Relations between two Schwartz kernels in dimensions $n$ and $n+1$
Let $(M,g)$ be an $n$-dimensional pseudo-Riemannian manifold and $\Box_g$ be the Laplace-Beltrami operator on $M$. Consider $z \in \mathbb{C}$ such that $\mathrm{ Im}(z)>0$, and we define $P_0 := \...
- 319
4
votes
1
answer
332
views
Exponentials and other functions of sums of anti-commuting operators
I know that if $A$ and $B$ are commuting operators, then $\exp(A+B) = \exp(A) \exp(B)$. Is there a similar formula if $A$ and $B$ are anti-commuting (that is, $AB+BA = 0$)?
I have developed a formula ...
3
votes
1
answer
127
views
The imaginary exponential of a tangent field on a manifold
If $M$ is a compact Riemannian manifold and $X$ is a tangent field, I am seeking to define the object $\exp {\mathrm i t X}$ for $t \in \mathbb R$, and I do not know how to do it.
One option was to ...
- 5,407
3
votes
0
answers
68
views
A strange convergence for a semigroup of operators
I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows:
Let $A,B$ ...
- 5,407
2
votes
0
answers
306
views
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 ...
- 7,609
1
vote
0
answers
77
views
A representation of positive matrix
Let $\mathcal H$ be a Hilbert space. Let $-\frac{1}{2}<r<0.$ Denote $c_p:=\int_{0}^\infty\frac{t^r}{1+t}dt.$ Suppose $A$ be a positive invertible operator in $B(\mathcal H).$ Is it true that $A^...
- 1,879
1
vote
0
answers
52
views
Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel
Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional
Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
- 705
0
votes
1
answer
203
views
For $B=\int \lambda d E_\lambda $ and $X$ commutes with every $E_\lambda $, why $BX$ is positive and self-adjoint?
Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E_\lambda $, then why ...
- 617
0
votes
2
answers
140
views
The derivative of a $C_0$-semigroup with respect to a perturbation parameter
Let $H$ be a Hilbert space, and $A : H \to H$ be the (semi-bounded) generator of the $1$-parameter $C_0$-semigroup $[0, \infty) \ni t \mapsto \mathrm e ^{-t A}$. Let $B : H \to H$ be a bounded ...
- 5,407
0
votes
0
answers
213
views
Convergence of inverse operator with projections
Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...
- 503