Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

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reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
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2answers
254 views

Unitary Operator as a complex valued function

A book on Quantum Mechanics by Schwinger states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator." Please give a hint on how to prove this assertion.
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166 views

Ascent and descent for a bounded linear operator

Let $T$ be a bounded linear operator on some complex Banach space. We define its ascent by $\alpha(T) = \min \{ n \ge 0 \, / \, N(T^n) = N(T^{n+1}) \}$ and its descent by $\delta(T) = \min \{ n \ge 0 ...
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311 views

Spectrum of a “quasi” right shift operator

Let $\mathcal{H}$ be a Hilbert space and let {$e_j$}$_{j\in \mathbb{Z}}$ be an orthonormal basis for $\mathcal{H}$. Define a linear operator $T$ on $\mathcal{H}$ by $T(e_0) = 0$ and $T(e_j) = e_{j+1}$ ...
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166 views

Densely-defined linear functionals and the spectrum of the adjoint operator

Let $L$ be a bounded linear operator acting on a complex Banach space $B$. If there exists a nonzero continuous linear functional $\ell \colon B \to \mathbb{C}$ such that $\ell(Lx)=\ell(x)$ for all $x ...
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140 views

Do the maximum and minimum values of a Laplacian eigenfunction have the same magnitude?

Let $\Delta$ be the scalar Laplace-Beltrami operator on a compact, connected, orientable 2-manifold without boundary smoothly embedded in $\mathbb{R}^3$ and let $\phi$ be one of its eigenfunctions, ...
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191 views

Construct a multiplication operator which has dense point spectrum

By a multiplication operator here we mean an operator $$Af(t)=m(t)f(t), \qquad f \in D(A)=\{x \in L^2(\mathbb{R} \mid m(t)f(t) \in L^2(\mathbb{R})\}$$ where $m$ is a Borel measurable function on ...
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614 views

Continuous spectrum can shrink to an isolated point

Let $A$ be a bounded linear operator in a Hilbert space $H$. I had the misconception that the continuous spectrum of $A$ would necessarily have some "continuous" appearance: an interval, a union of ...
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99 views

Does a projection valued measure (PVM) induce a PVM on a generic subspace of the Hilbert space?

Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of ...
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265 views

How to characterize self-adjoint operators in terms of orthogonal diagonalizability

Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I): The fundamental quality required of operators representing physical quantities in ...
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Changing the manifold, preserving the discrete spectrum

On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. If $M$ is not compact, then $L$ admits a continuous spectrum. Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
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When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
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1answer
272 views

Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?

The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be ...
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272 views

Convergence of spectra under strong convergence of operators

Say $\left\{A_n\right\}$ is a sequence of bounded self-adjoint operators on a separable Hilbert space, converging in strong operator topology to a (bounded, self-adjoint) operator $A$. Denote the ...
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694 views

spectrum of right shift operator on $\ell^2(\mathbb{Z})$

Consider the right shift operator on $\ell^2(\mathbb{Z})$. Is there a way of calculating (well, showing what it is since I already know it's $z$ s.t $|z| = 1$) its spectrum without reference to it ...
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1answer
536 views

Spectrum of sum of operators on Banach spaces

Let $A$ and $B$ be two operators on a Banach space $X$. I am interested in the relationship between the spectra of $A$, $B$ and $A+B$. In particular, are there any set theoretic inclusions or ...
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2answers
613 views

How do the solutions to the wave and heat equations converge in general?

I would like to check my understanding with someone if possible. When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
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Example of a compact set that isn't the spectrum of an operator

This question is a follow-up to this recent question and related to that one. Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset ...
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Can spectrum “specify” an operator?

Given a bounded operator $A$ on a Banach space $X$, one may find the spectrum $\sigma(A)\subset{\bf C}$. Here are my questions: Given some set in the complex plane, say, $S\subset{\bf C}$, ...
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1answer
197 views

Schrödinger operator: where is the generator to be defined?

The theory as I know it Let $\mathcal{H}$ be a Hilbert space and $(A, D(A))$ a self-adjoint operator acting on it. The Spectral Theorem (cfr. Reed & Simon Methods of modern mathematical physics, ...
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Spectral decomposition of a normal matrix

I'd like to find the spectral decomposition of $A$: $$A = \begin{pmatrix} 2-i & -1 & 0\\ -1 & 1-i & 1\\ 0 & 1 & 2-i \end{pmatrix}$$ i.e. $A=\sum_{i}\lambda_i P_i$ where ...
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references for the spectral theorem

Recently, I am thinking about the question in spectral theory. And I finally found that I need help with the properties of unitary operator. Its a consequence of the spectral theorem for the normal ...
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628 views

spectrum of the “discrete Laplacian operator”

In numerical analysis, the discrete Laplacian operator $\triangle$ on $\ell^2({\bf Z})$ can be written in terms of the shift operator $\triangle=S+S^*-2I$ where $S$ is the right shift operator. ...
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spectrum of right shift operator

Here is the question: Considering the right shift operator $S$ on $l^2({\bf Z})$, what can one know about ran$(S-\lambda)$? Here is what I thought: If one wants to prove that the operator ...
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Matrix Differential Equation with a Skew-Symmetric Matrix

From a bank of masters exams: Say the position of a particle moving in $\mathbb{R}^n$ is given by a smooth vector-valued function $\vec{x}(t)$. Suppose that $\vec{x}(t)$ satisfies a ...
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Spectrum of a linear operator

Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set: $$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{z}}$$ for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
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Basic Spectral Theory Problem: Finding the Point/Continuous Spectrum of an Operator

I have the following problem: Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for ...
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1answer
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What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
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1answer
917 views

Spectrum of a product of operators on a Banach space

Let $A$ and $B$ be two operators on a Banach space X. I am interested in the relationship between the spectra of $A$, $B$ and $AB$. In particular, are there any set theoretic inclusions or everything ...