All Questions
Tagged with functional-calculus fa.functional-analysis
27 questions
10
votes
1
answer
695
views
Rigorous proof of the pentagon identity
I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...
- 1,391
10
votes
0
answers
657
views
“Taylor series” is to “Volterra series” as “Laurent series” is to _________?
Preamble
My question is similar to an earlier MathOverflow question:
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
- 446
7
votes
1
answer
184
views
Functional calculus on the Schwartz space instead of $L^2$?
As far as I know, functional calculus is typically carried out on Hilbert spaces with (possibly unbounded) self-adjoint operators.
However, I wonder if there is a way to do it on the space of test ...
- 3,487
6
votes
3
answers
603
views
Differential calculus of functions of self-adjoint operators
Let $H$ be a Hilbert space over $\mathbb{C}$. Fix a self-adjoint operator $A:D(A)\rightarrow H$ and a Borel function $f:\mathbb{R}\rightarrow\mathbb{C}$. The operator $f(A)$ is defined by the spectral ...
- 61
6
votes
1
answer
1k
views
Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
- 969
5
votes
1
answer
324
views
Hölder continuity of functional calculus
Let $0<\beta<1$ and $ f \colon [0,1] \to [0,1]$ be $\beta$ Hölder continuous with constant $C$. Let $H$ be a Hilbert space and $A,B$ be self adjoint operators on $H$, such that $\sigma(A+B),\...
- 475
4
votes
1
answer
428
views
Reference Request: Calculus of Variations in Hilbert Space
I'm looking for a good reference to a book on calculus of variations in the setting of Banach Spaces.
If it helps, I'm working with a particular functional acting on Fr\'{e}chet-differentiable ...
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
1
answer
104
views
From Wightman to HK axioms for "non-neutral (charged?)" fields
Wightman axioms deal with operator-valued distributions (Wightman fields) whose values are unbounded operators in general.
On the other hand, the Haag-Kastler axioms deal with net of observables, ...
- 3,487
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
2
answers
536
views
Hilbert Scale Inclusions
I'm looking at properties of the scale of Hilbert spaces $(X_s)_{s\in \mathbb{R}}$, which are constructed as follows. Starting with $A:D(A)\subset H\to H$, $A$ a densely defined, strictly positive ($...
- 379
2
votes
1
answer
521
views
Fréchet derivative of evaluation-like functional (multivariate)
I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following.
Let $H$ be ...
- 5,405
2
votes
0
answers
156
views
What would be the explicit formula for the remainder in Taylor's theorem for functional calculus? [closed]
Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function and $A,B$ be $n \times n$ self-adjoint matrices that commute.
Then, I see that $f(A+tB)$ is a well-defined matrix-valued function for real ...
- 3,487
2
votes
0
answers
137
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
2
votes
0
answers
122
views
Reference on iterated integrals against projection valued measures
I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which ...
- 183
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
1
answer
758
views
How to compute integral of a gaussian over a noncentered ball?
Let $\mathcal{B}(x,r)$ the ball of center $x \in \mathbb{R}^n$ and radius $r>0$ (so $\mathcal{B}(x,r) = \{y \in \mathbb{R}^n : \|y-x\| \leq r\}$, where all norms are $\ell^2$-norms).
I would like ...
- 21
1
vote
0
answers
170
views
Generalization of Borel functional calculus
[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus]
Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
- 95
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
1
vote
0
answers
64
views
Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?
Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.
Then, they are simultaneously ...
- 3,487
1
vote
0
answers
135
views
infinite dimensional funtional ito calculus
I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
- 5,405
0
votes
1
answer
381
views
Converse of Lax-Milgram theorem [closed]
Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V.
Assume that, for any continuous linear functional on $l \in V’...
- 23
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
0
votes
0
answers
230
views
A gap in the proof of uniqueness of functional calculus based on a spectral theorem
This question considers the proof of a fundamental theorem of functional calculus, given in the book Spectral Theory -
Basic Concepts and Applications by David Borthwick (Theorem 5.9).
Firstly we have ...
- 1,755
0
votes
0
answers
62
views
"Trade-off" between bound on the function and on the spectrum for functional calculus in spectral theory
Let $A$ be a self-adjoint (unbounded) operator on a separable Hilbert space $H$.
From the following form of spectral theorem, we may define a functional calculus by $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
- 1,755