Questions tagged [functional-calculus]
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52 questions
0 votes
0 answers
347 views
$A$ self-adjoint then $A^2$ self-adjoint?
Let $H$ be a Hilbert space. Let $A:D(A)\to H$ be a self-adjoint operator. When can we say $A^2:D(A^2) \to H$ is self-adjoint? Is it enough to define $D(A^2)=\bigl\{u\in D(A) | Au\in D(A)\bigr\}$? And ...
- 181
0 votes
0 answers
170 views
Boundedness of operator in $L^p$ space for $p\neq 2$
This problem arises when I'm reading this paper about Internal modes for quadratic Klein-Gordon equation in $\mathbb R^3$, written by Tristan Léger and Fabio Pusateri: https://arxiv.org/abs/2112.13163....
- 101
2 votes
1 answer
211 views
Functional derivative/Gateaux derivative of functions of probability densities
Let $(\Omega, \mathcal{A})$ be a measure space and $\mathcal{P}$ be a convex set of probability distribution on that space. Furthermore let $\mu$ be a $\sigma$-finite measure that dominates every ...
- 155
1 vote
0 answers
119 views
Norm of the resolvent of the Laplacian acting on $L^p$
Let $M$ be a closed Riemannian manifold and let $\Delta$ be the positive Laplacian acting on functions on $M$. If $\lambda$ is not an eigenvalue of $\Delta$, we have the resolvent operator $R_\lambda=(...
- 374
1 vote
0 answers
188 views
Modern reference for contour integral representation of projection operators?
I am currently typing something up and a technical part of the argument requires me to show that convergence of a sequence of operators in resolvent norm implies convergence of the eigenvalues. I ...
- 222
1 vote
0 answers
108 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^...
- 1,970
3 votes
1 answer
170 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, ...
- 3,745
5 votes
0 answers
269 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 := \...
- 339
7 votes
1 answer
299 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 ...
- 3,745
2 votes
0 answers
220 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 ...
- 115
2 votes
1 answer
569 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\...
- 21
0 votes
0 answers
254 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 ...
- 537
10 votes
1 answer
867 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 ...
- 1,583
1 vote
0 answers
79 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 ...
- 720
5 votes
0 answers
169 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
200 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 ...
- 3,745
4 votes
1 answer
451 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 ...
1 vote
0 answers
92 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 ...
- 3,745
10 votes
0 answers
742 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
8 votes
2 answers
700 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
2 votes
4 answers
445 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 ...
- 61
3 votes
0 answers
241 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]}{\...
- 31
0 votes
0 answers
261 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 ...
- 1,785
0 votes
0 answers
77 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 $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
- 1,785
5 votes
1 answer
405 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),\...
- 485
2 votes
0 answers
137 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 ...
- 5,009
2 votes
1 answer
680 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 ...
- 4,942
4 votes
0 answers
219 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 ...
- 747
3 votes
0 answers
93 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 $A,B$ ...
- 5,477
0 votes
2 answers
241 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 ...
- 5,477
3 votes
1 answer
889 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 ...
- 41
1 vote
1 answer
316 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-...
- 223
2 votes
2 answers
618 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
14 votes
1 answer
1k 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 ...
- 5,005
2 votes
0 answers
174 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 ...
- 839
2 votes
0 answers
192 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 ...
- 193
3 votes
1 answer
155 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 ...
- 5,477
1 vote
0 answers
105 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? $$ I_1\equiv\int \frac{1}{|\Sigma|^a|\Xi|^b|\mathrm{L}\Sigma\...
- 11
2 votes
0 answers
365 views
Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum
I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
- 7,781
6 votes
1 answer
1k views
Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
- 969
0 votes
1 answer
314 views
Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]
In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important: $$\bigtriangleup F(x):=F(x+1)-...
- 109
0 votes
1 answer
248 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 ...
- 135
0 votes
1 answer
236 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 ...
- 663
0 votes
1 answer
241 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 ...
- 111
3 votes
1 answer
414 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 ...
- 146
0 votes
1 answer
416 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
1 vote
0 answers
172 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 ...
- 151
6 votes
3 answers
788 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
1 vote
1 answer
229 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, ...
- 4,496
4 votes
1 answer
485 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 ...
