# Questions tagged [functional-calculus]

Functional calculus allows the evaluation of a function applied to a linear operator or a matrix. The function could be a polynomial, a holomorphic function, a continuous function or a measurable function defined on the spectrum of an operator or a Banach algebra. Functional calculus is a basic and powerful tool in the spectral theory of operators and operator algebras and is part of functional analysis.

273 questions
Filter by
Sorted by
Tagged with
41 views

48 views

### $U$ unitary: $\mathbb{T}\ne\sigma(U)$. Prove $\forall\varepsilon>0$ there exists a polynomial $p(z)$ such that $\|U^{-1}-p(U)\|<\varepsilon.$

Let $U$ be a unitary operator: $\mathbb{T}=\{\lambda:|\lambda|=1\}\setminus\sigma(U)\ne\varnothing$ (the spectrum does not cover the whole circle). Prove that $\forall\varepsilon>0$ there exists a ...
40 views

10 views

### Differential of fraction of function involve basis spline

Suppose, we want to find $\dfrac{\partial X_i(t)}{\partial c}$ where $X_i(t)=\dfrac{exp[Z_i(t)}{\int_a^dexp[z_i(s)]ds}$, $i=1,...,n$ and $Z(t)=b(t)^Tc + b(t)^TAU.$ Here $b(t)=[b_1(t),...b_p(t)]$ be a ...
21 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: $\frac{\partial J}{\partial y}|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\Delta\sigma}$,...
37 views

### Matrix differentials to estimate the critical value from an equation that involves double summation and integration

The equation I have, $l(c)=-\sum_{i=1}^n\sum_{j=1}^{m_i}\dfrac{||y_{ij}-x_i(t_{ij})||^2}{2\sigma_e^2}$ where $x(t)=\dfrac{exp[b(t)^Tc+b(t)^TAu]}{\int_a^vexp[b(s)^Tc+b(s)^TAu]ds}$ Theoretically s or t ...
44 views

48 views

### Notational confusion in derivation of Euler-Lagrange equations

I'm reading Bishop's "Pattern Recognition and Machine Learning" section on the Calculus of Variations (Appendix D) and he defines the functional derivative of $\frac{\delta F}{\delta y(x)}$ ...
35 views

### Trying to derive the path integral from first principles (step 1)

I was recently told in the physics forum (https://physics.stackexchange.com/questions/616186/deriving-the-path-integral-from-the-time-slice-approach-for-a-general-hamiltonia) that it is not possible ...
45 views

### functional calculus on C*-algebra and majoration of $||f(A+B)-f(A)||$

In introductory books on C*-algebra, functionnal calculus is quicly presented as it is a powerful tool in the field. It is for example prooved that for an operator $a$ and a function $f$ continous on ...
56 views

28 views

40 views

### Chain rule for functional derivatives

I'm studying the functional derivatives on Mahler's book "Statistical Multisource-Multitarget Information Fusion". I have some difficulties in understand a chain rule for functional ...
171 views

59 views

### Calculate the Euler-Lagrange for a functional with two nested integrals?

I've been reading papers about a fairly unknown topic in quantum mechanics called the quantum backflow effect. And in many of the papers they find an eigen value problem corresponding to the maximal ...
38 views

### How to prove $f(A^*) =f(A)^*$

I'm studying Conway's functional calculus. If $\mathcal{X}$ is banach space and $A \in \mathcal{B}(\mathcal{X})$, and $f \in \mathtt{Hol}(A)$ , show that $f(A)^* = f(A^*)$ Firstly, I tried it for ...
168 views

### Functional Derivative with Discrete Variable

Problem $$\text{Find}\quad\frac{\delta F_k}{\delta G} \quad \text{given} \quad F_k=\left(\sum_{r=0}^{N-1} e^{ikr}\int_{-\infty}^{\infty} dt \ e^{i\omega t} G(r,t)\right)^{-1}$$ noting that $k$ and $r$ ...
46 views

### An inequality of positive maps of roots of positive operators on C*-algebras

This is a pretty long question that I have been struggling with for a while, any tips or suggestions would be awesome! Let $A, B$ be two C*-algebras, the later algebra is represented on a complex ...
29 views

### Understanding $z(a,b) \in[0,1]$ where $a=|x-y|$, $b=|y|$ and $x,y\in[-2,2]$

I previously asked a poorly written question that used incorrect terminology. A user @epiliam was very patient and worked me through my logic to understand what I was trying to ask. I had a question ...
I am learning the theory of standard von Neumann algebras by following Chapter $10$ from the book 'Lectures on von Neumann algebras' by Strătilă and Zsidó. But I got stuck with an argument provided in ...