Articles of functional calculus

Neumann series expansion for the resolvent

If $T$ is an operator on $l_2$, and $\lambda>r(T)$ (where $r(T)$ denoted the spectral radius of $T$), the resolvent $(\lambda I-T)^{-1}$ can be expanded as $$ (\lambda I-T)^{-1}=\frac{1}{\lambda}I+\frac{1}{\lambda^2}T+\frac{1}{\lambda^3}T^2+\dots $$ If we fix some $x\in l_2$, and denote $y:=(\lambda I-T)^{-1}x$ the above expansion implies that $y$ is in the closed span of $(T^nx)_{n=0}^{\infty}$. Is $y$ in […]

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 $\mathbb{R}$, then we have necessarily $$\inf_{t\in \mathbb{R}}|e^{tA}x|>0 \ \ \ \ \ or \ \ \ \ e^{tA}x=0 \ […]

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 the characteristic polynomial of $A$ then $f(A) = 0$. (This exercise was taken from […]

almost unitaries are close to a unitary element

I need help to prove the following exercise: Let $\epsilon >0$. Show that there exists $\delta >0$ with the property: If $A$ is a unital $C^*$-algebra and $x\in A$ such that $\|x^*x-1\|<\delta,\;\|xx^*-1\|<\delta$, then there is a unitary element $u\in A$ with $\|x-u\|<\epsilon$. To prove this, I use continuous functional calculus for the element $x^*x$ and […]

Functional Derivative ${\delta q_a(t)}/{\delta q_b(t')}$

$\newcommand{\fdv}[2]{\frac{\delta #1}{\delta #2}}$ $\newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d}#2}}$ $\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}$ I’m from a physics background and I’ve always known the definition to be related to the Euler-Lagrange Equations i.e. $$\fdv {L(q,\dot q)} {q(t)} = \pdv{L}{q} – \dv{}{t} \pdv{L}{\dot q} \; ,$$ where $\dot q$ denotes the derivative of the function $q$ with respect to $t$. However with […]

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the product is the composition of operators. If $T,F:(0,\infty)\to\mathcal{L}(X)$ are differentiable functions, can we apply the product rule to derive $T(t)F(t)$? Particularly, I’m interested in the case that $\{T(t)\}_{t\geq […]