Articles of spectral theory

Complex hermite polynomials as an eigenfunction of an operator?

The real Hermite polynomials are given by $$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$ for $x\in \mathbb R$ and $n=0, 1, 2 …$. The Hermite function $H_n$ of order $n$ is an eigenfunction of the harmonic oscillator $\Delta=-\frac{\partial^2}{\partial x^2}+x^2$ corresponding to the eigenvalue $2n+1$, i.e., $$\Delta H_n (x)=(2n+1) H_n(x) .$$ I would like to know, what happens for the complex […]

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix $A$? Thank you very much!

irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book “A short course on spectral theory“, William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be irreducible if the only closed $\pi(A)$-invariant subspaces of $H$ are the trivial ones $\{0\}$ and $H$. Show that $\pi$ is irreducible […]

Spectrum of left shift operator $L\in B(H)$

Let $H$ be a Hilbert space with an orthonormal base $e_i$ and $L$ the left shift operator $L\in B(H)$: $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$. I computed the spectrum could someone please tell me if this is right? My work: $\lambda \in \sigma (L)$ if and only if there exists $x \in H$ ($x\neq […]

Spectral Measures: Spectral Spaces (I)

Problem Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its probability measures by: $$\nu_\varphi(A):=\|E(A)\varphi\|^2$$ Introduce the spectral space: $$\mathcal{H}_\parallel:=\{\varphi:\nu_\varphi\ll\lambda\}$$ $$\mathcal{H}_\perp:=\{\varphi:\nu_\varphi\perp\lambda\}$$ Then they decompose: $$\mathcal{H}=\mathcal{H}_\parallel\oplus\mathcal{H}_\perp$$ How to prove this? Attention This thread has been split: Spectral Spaces (II)

Perturbation by bounded operators

Let $H$ be a complex Hilbert space. Denote $\mathcal{L}(H)$ the space of linear bounded operators on $H$ into $H.$ Let $N \in \mathcal{L}(H) $ be normal ($N^* N =NN^*$) such that $\sigma(N) \subset \Gamma := \{z \in \mathbb{C}\backslash \{0\}, -\pi /2<Arg(z) <\pi/2 \}.$ It is clear that there exists $C>0$ such that \begin{equation}\tag{1} (N+\lambda)^{-1} \in […]

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 K(x,t)y(t)dt=\sqrt{2x^2-2x+1}$$ where $$K(x,t)=\max((1-x)t,(1-t)x),0<x<1,0<t<1.$$ Eigenfunctions of $ K(x,t)$ was found by @oenamen in the answer to the above-mentioned question. I thought one should be able to […]

Example: Operator with empty spectrum

I tried Google and a few books but couldn’t find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your help!

A simple question about *-homomorphism in C*-algebra

Let $A$ and $B$ be C*-algebra, $h\colon A\rightarrow B$ is *-homomorphism. If $a\in A_{\operatorname{sa}}$, then $\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. Here, $A_{\operatorname{sa}}$ denotes all the self adjoint elements in $A$ How to prove this inclusion?

Spectral radius, and a curious equality.

Given a $N\times N$ matrix $A$ over $\mathbb R$. Let $ \rho\left( A \right) = \max \left\{ {\left| \lambda \right|;\lambda \mbox{ eigenvalue of }A} \right\}$. Someone told me that, the following holds: $$ \rho\left( A \right) = \inf \left\{\lVert| A \rVert|,\lVert \cdot \rVert\mbox{ matrix norm}\right\} . $$ Here is the definition of a matrix norm […]