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.

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Cauchy's integral formula for Cayley-Hamilton Theorem

I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise: Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ matrix over $\mathbb C$ and $f(z) = \det(z-A)$ is…
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Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first definition is based on zeta function…
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Functional differential equation (from Quantum Field Theory).

I have a certain differential equation that includes functional derivatives. I know the solution, but I'm having a hard time to show that the equation is indeed solved by the solution. The background for this question is quantum field theory (in…
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A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the function $t\mapsto e^{tA}x$ is bounded on…
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In the Physicists' definition of the path integral, does the result depend on the choice of partitions?

The standard definition of the path integral in Quantum Mechanics usually goes as follows: Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,t_n\}$$ with $t_k = t_0 + k\epsilon$ where…
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Smoothness of $O(n)$-equivariant maps of positive-definite matrices

$\def\sp{\mathrm{Sym}^+}$Let $\sp \subset GL(n,\mathbb R)$ denote the manifold of positive-definite symmetric $n \times n$ matrices. I am interested in functions $A : \sp \to \sp$ that are equivariant under the natural conjugation action of $O(n)$;…
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Searching two matrix A and B, such that exp(A+B)=exp(A)exp(B) but AB is not equal to BA.

We know that if two matrix $A$ and $B$ commutes then $\exp(A+B)=\exp(A)\exp(B)$. I am trying to find two matrix that does not commute but $\exp(A+B)=\exp(A)\exp(B)$ is true for them. Can anybody give exact example. Thanks
Fin8ish
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When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in $f_0$, the $f$ that minimizes $J$. Solutions are…
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Functional derivatives in (Physics) Field Theory

The functional or variational derivative as defined in several places like Wikipedia seems to be defined as a functional, $L$ that takes a single input function, say $f(x)$ and then we define a certain object $$\frac{\delta L}{\delta f(x)}$$ that is…
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Is there some approach to make functional integrals rigorous?

Quantum Mechanics and Quantum Field Theory can both be formulated in terms of the so-called functional integrals. The point is that intuitively it is an "integral over all possible paths" or rather "integral over all possible field configurations",…
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Exponential of the Laplacian operator as diffusion equation

Let $u$ be a function on a domain $\Omega$ with some fixed boundary condition. I have recently seen a notation $e^{\tau \Delta}u$ as meaning the the time evolution of $u$ by diffusion for a time $\tau$. I'm curious where this notation comes from,…
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If an element in a Banach algebra is anihilated by an analytic function then it must be algebraic.

Let $A$ be a Banach algebra, let $a\in A$ and suppose $f(a)=0$, where $f$ is an analytic function defined on an open set $U$ containing $\sigma(a)$. Prove that $a$ is algebraic in the sense that $p(a)=0$ for some polynomial $p$. PS: I have just…
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Proof that a minimum problem has no solution

Please I have an exam in a few days, can you help me with the following exercise? Let $A=\{x\in\mathbb{R}^2: 1<|x|<2\}$ and $M\geqslant 0$. On the set $\mathcal{A}_{M}=\{u\in C(\bar{A})\cap C^1(A):u=0 \text{ on } |x| \text{ and } u=M \text{ on }…
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Searching for a proof that in a normed functional space of $C^0[0,1]$ with sup norm, that norm is nowhere differentiable.

Having a normed linear space $S=C^0[0,1]$ of continuous functions $f:[0,1] \rightarrow \Bbb R%$, with sup norm: $\|f\|=\sup_{\space x \in [0,1]}|f(x)|$, prove that $F(f)=\|f\|$ is nowhere differentiable in $S$, that is, for all $f_0 \in S$, there…
Dagon
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Strong continuity of the Borel functional calculus

I have sometimes heard that the Borel functional calculus maps bounded pointwise convergent sequences of Borel functions to strongly convergent sequences of operators. I gather "sequence" is important here, due to the measure theory aspect, we…
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