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Questions tagged [functional-calculus]

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2
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0answers
95 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 ...
5
votes
1answer
128 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
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1answer
201 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
1answer
86 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
1answer
170 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
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1answer
79 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
1answer
188 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 ...
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0answers
75 views

Question about holomorphic functional calculus

I just try to understand sectorial operators and the holomorphic functional calculus. For that I work with some example, namely the multiplication operator. The general setting: let $X$ be a metric ...
0
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1answer
137 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
0answers
83 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
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3answers
331 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
1answer
156 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
1answer
221 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
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0answers
101 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
1answer
156 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 - ...