# Questions tagged [functional-calculus]

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### 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 $\mathcal{H}$, ...
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### 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 ...
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### 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)-...
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### 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 ...
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### 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 ...
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### 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 ...
270 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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### 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 ...
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 - ...