Questions tagged [functional-calculus]
The functional-calculus tag has no usage guidance.
44
questions
1
vote
0
answers
145
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 ...
0
votes
0
answers
32
views
Resolvent estimates for Laplacians on directed graphs
Let us consider a directed and weighted graph $G$ with $N = |G|$ nodes. Denote the corresponding (weighted) adjacency matrix by $W \in \mathbb{R}^{N\times N}_{\geq 0}$ and let $D$ be the diagonal in-...
2
votes
1
answer
291
views
How to solve an optimization problem whose optimization variable is a function?
I would like to find an optimal probability density function (PDF) $f$. Given $b$,
$$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\...
0
votes
0
answers
192
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 ...
10
votes
1
answer
632
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
vote
0
answers
49
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 ...
5
votes
0
answers
130
views
Functional inverse problem based on a variational principle
I am trying to solve an inverse problem based on variational principle.
I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently ...
2
votes
0
answers
145
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
votes
1
answer
269
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 ...
0
votes
0
answers
42
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 ...
10
votes
0
answers
601
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 ...
8
votes
2
answers
601
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}$, ...
2
votes
4
answers
314
views
EM-wave equation in matter from Lagrangian
Note
I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success.
Setup
Let's suppose a ...
3
votes
0
answers
193
views
Gelfand "Calculus of Variation" 1.7 question on definition and purpose of variational derivative
In Gelfand Calculus of Variation, chapter 1.7, the variational derivative is defined as:
$$\left.\frac{\partial J}{\partial y}\right|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\...
0
votes
0
answers
215
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 ...
0
votes
0
answers
61
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 $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
5
votes
1
answer
296
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),\...
2
votes
0
answers
111
views
Can all (inverse) trigonometric functions with periodic iterates be characterized?
I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...
1
vote
1
answer
405
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 ...
4
votes
0
answers
157
views
Pseudodifferential Operators and Functional Calculus
I hope this is not too naive a question for MO. I've been taking a mathematical physics course, and was shown how operators like $\sqrt{1-\Delta}$ could be defined by taking multiplication operators ...
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 $A,B$ ...
0
votes
2
answers
126
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 ...
1
vote
1
answer
656
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 ...
1
vote
1
answer
209
views
Variational problem: how to minimise the second moment?
This is a neater version of a question I posted here, on which I'm also stuck.
The problem: Say I have a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-...
2
votes
2
answers
483
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 ($...
13
votes
1
answer
793
views
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
Volterra ...
2
votes
0
answers
131
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 ...
2
votes
0
answers
100
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 ...
3
votes
1
answer
123
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 ...
1
vote
0
answers
87
views
Condition for the integrability of a matrix function
Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable?
$$
I_1\equiv\int \frac{1}{|\Sigma|^a|\Xi|^b|\mathrm{L}\Sigma\...
2
votes
0
answers
278
views
Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum
I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
6
votes
1
answer
983
views
Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
0
votes
1
answer
266
views
Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]
In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-...
0
votes
1
answer
178
views
Does Borel functional calculus commute with *-isomorphism?
I am confused with the underlined equation in the following picture.
I know that a *-isomorphism commutes with continuous functional calculus since every continuous functions on the compact subset of ...
0
votes
1
answer
202
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 ...
0
votes
1
answer
186
views
Variation in Einstein-Hilbert action [closed]
In this page there are calculations of variation of Einstein-Hilbert action.
I see variations of terms like this:
$\delta {R^{\rho }}_{{\sigma \mu \nu }}$
where the term is not a functional, and ...
3
votes
1
answer
330
views
Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth Functions?
I'm not an analyst, so forgive me if what I'm asking is not suitable for Mathoverflow.
For convenience, let $X$ be a compact complex manifold, and $E$ a holomorphic vector bundle on $X$. Let $H$ be ...
0
votes
1
answer
348
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’...
1
vote
0
answers
122
views
Sufficient condition for gradient existence in Hilbert spaces
Let $\mathbb H$ a Hilbert space and $N:\mathbb H\to \mathbb H$ a continuous nonlinear mapping. In Fonda and Mawhin (Iterative and variational methods for the solvability of some semilinear equations ...
6
votes
3
answers
538
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 ...
1
vote
1
answer
186
views
Optimal joint coupling of all probability measures on a 3 point space
I am looking for any remotely related reference for the following problem, for which I have not the least clue what techniques would be useful.
Consider a discrete probability space $\Omega = \{x, y, ...
4
votes
1
answer
400
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 ...
1
vote
0
answers
132
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 ...
3
votes
1
answer
323
views
Integrating the resolvent of a self-adjoint operator across a continuous part of the spectrum
Let $A$ be a closed self-adjoint operator on a Hilbert space $H$, possibly unbounded and hence defined on a dense domain $D(A) \subset H$. It is well known that integrating the resolvent $R_z = (z I - ...