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.
Questions tagged [functional-calculus]
342 questions
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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…
Sam
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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…
anon
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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…
AccidentalFourierTransform
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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…
user165633
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The matrix logarithm is well-defined - but how can we *algebraically* see that it is inverse to the exponential, as a finite polynomial?
This question is inspired by this which I saw earlier today. I started writing my answer, to share the insight that the matrix logarithm can be defined on matrices that do not have unit norm using an alternative technique.
Now, Sangchul has posted a…
FShrike
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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…
Gold
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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 $O(n)$;…
Anthony Carapetis
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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…
Chay Paterson
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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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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,…
rviertel
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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…
guillefix
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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",…
Gold
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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…
Jeff
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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…
Ruy
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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 }…
Pefok
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