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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 ...
oggius's user avatar
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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-...
Qualearn's user avatar
2 votes
1 answer
309 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\...
Erik's user avatar
  • 21
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0 answers
200 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 ...
John's user avatar
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10 votes
1 answer
637 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 ...
Estwald's user avatar
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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 ...
sorrymaker's user avatar
5 votes
0 answers
132 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 ...
can't stop me now's user avatar
2 votes
0 answers
147 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 ...
Isaac's user avatar
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3 votes
1 answer
285 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 ...
Stephen Montgomery-Smith's user avatar
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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 ...
Isaac's user avatar
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10 votes
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610 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 ...
Nike Dattani's user avatar
8 votes
2 answers
609 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 $\mathcal{H}$, ...
Ryszard Kostecki's user avatar
2 votes
4 answers
318 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 ...
Chopin's user avatar
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198 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]}{\...
MkSn1999's user avatar
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0 answers
218 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 ...
Ma Joad's user avatar
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"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 $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
Ma Joad's user avatar
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5 votes
1 answer
302 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),\...
Paul Pfeiffer's user avatar
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 ...
Max Muller's user avatar
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2 votes
1 answer
430 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 ...
ABIM's user avatar
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4 votes
0 answers
160 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 ...
Wahome's user avatar
  • 737
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 $A,B$ ...
Alex M.'s user avatar
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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 ...
Alex M.'s user avatar
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1 vote
1 answer
689 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 ...
fluxy44's user avatar
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1 vote
1 answer
219 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-...
user655870's user avatar
2 votes
2 answers
494 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 ($...
user2379888's user avatar
13 votes
1 answer
820 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 ...
Mike Battaglia's user avatar
2 votes
0 answers
134 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 ...
mohi's user avatar
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2 votes
0 answers
105 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 ...
Cabbage's user avatar
  • 183
3 votes
1 answer
124 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 ...
Alex M.'s user avatar
  • 5,335
1 vote
0 answers
88 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\...
cong's user avatar
  • 11
2 votes
0 answers
282 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 ...
Hugo Chapdelaine's user avatar
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 ...
Dave Shulman's user avatar
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)-...
soodehMehboodi's user avatar
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 ...
C. Ding's user avatar
  • 135
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 ...
user92646's user avatar
  • 617
0 votes
1 answer
188 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 ...
asv's user avatar
  • 111
3 votes
1 answer
339 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 ...
Kashayar's user avatar
  • 146
0 votes
1 answer
351 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’...
Simon Pun's user avatar
1 vote
0 answers
123 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 ...
Arturo Sanjuán's user avatar
6 votes
3 answers
549 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 ...
Chuwei Zhang's user avatar
1 vote
1 answer
187 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, ...
John Jiang's user avatar
  • 4,354
4 votes
1 answer
405 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 ...
Rombaldo Meniscus's user avatar
1 vote
0 answers
133 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 ...
ABIM's user avatar
  • 4,775
3 votes
1 answer
325 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 - ...
Igor Khavkine's user avatar