# Tagged Questions

Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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### Spectrum of shift-operator

Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ...
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### Relation of Hodge Theorem to Eigenfunction Basis of Laplacian

The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of ...
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### Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
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### Spectral radius as the inf of norms of conjugates

I need help with the following problem: Let $A$ be a unital $C^{*}$-algebra. (a) If $r(a)<1$ and $b=(\sum_{n=0}^{\infty}a^{*n}a^{n})^{1/2}$, show that $b\geq 1$ and $||bab^{-1}||<1$. (b) ...
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### Resolvent properties

Suppose that $A$ is a $n \times n$ matrix with $n$ different eigenvalues $\lambda_k.$ Corresponding eigenvectors are denoted as $x_k$, $x_k^Tx_k =1.$ Now $A=X\Lambda X^{-1}$. Denote $Q=X^{-1}$. ...
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### Spectrum of a product

Let $A$ be a unital $C^{*}$-algebra. I am trying to show that if $a,b\in A$ are positive elements, then the spectrum of $ab$ is contained in the positive real numbers. I know that in the commutative ...
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### Inverse of bounded self adjoint operator on HS is self adjoint?

Let $A=A^{*}$ be a bounded self adjoint operator on a Hilbert space $\mathcal{H}$ with Range Ran$(A) = D$ dense in $\mathcal{H}$. $A$ is injective, since Ran$(B) \perp ker(B^{*}) = ker(B)$. So ...
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### Eigenvalues and adjoint of operator $T(x_k)_{k=1}^{\infty} = (x_{2k})_{k=1}^{\infty}$

Let $T$: $l^2 \rightarrow l^2$ denote the operator \begin{align} T(x_1,x_2,\dots, x_n,\dots) = (x_2,x_4,\dots,x_{2n},\dots). \end{align} There are several questions regarding this operator that I need ...
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### Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
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### Square root in Banach algebra

Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin ...
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### The eigenvectors of a matrix and its transpose that correspond to the same eigenvalue are not orthogonal

Spent hours trying to prove this after encountering it in Lax's discussion of the spectral theorem, but no luck. Here's the problem (it is Theorem 18 in Lax 2ed, Chapter 6): A mapping $A$ has ...
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### Spectrum of Symmetrizable Matrix

A matrix $M$ is symmetrizable if $M = D S$ with $D$ a square diagonal matrix with positive entries, and $S$ a symmetric matrix. What can be said about the spectrum of $M$? It seems like I ...
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### Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
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### Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
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### Asymptotic of the heat kernel

I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian on a Riemannian Manifold" and not quite clear how to get the estimate $(4\pi t)^{n/2}|Q_k * H_k|\leq C \cdot t^{k+1}$ on a compact ...
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### An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...