Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.
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4answers
92 views
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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vote
1answer
58 views
A question on strongly continuous semi-groups
At the moment I am trying to understand "Lectures on Floer homology" By D. Salamon, see
http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf
In step 1 of the proof of Lemma 2.4 (page 17) he ...
1
vote
1answer
82 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 ...
3
votes
1answer
233 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 $T(e_j) = e_{j+1}$ ...
3
votes
1answer
114 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 ...
3
votes
1answer
89 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, ...
3
votes
0answers
111 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 ...
3
votes
0answers
395 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 ...
4
votes
2answers
140 views
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 $M'$ ...
4
votes
0answers
133 views
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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vote
0answers
93 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 ...
2
votes
1answer
181 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 ...
12
votes
1answer
621 views
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 ...
7
votes
1answer
218 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 ...
40
votes
1answer
1k views
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 ...
5
votes
1answer
163 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 ...
0
votes
1answer
535 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 ...
1
vote
1answer
296 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 ...
0
votes
0answers
121 views
Uniqueness of solution to a differential equation
Consider the second-order ODE
$-(py')'+(q-\lambda$$w)y=wf$ $(1)$
with $y$ an $L^2$ complex-valued function on $[a,b]$ subject to the boundary conditions:
...
2
votes
1answer
150 views
Schrödinger operator: where is the generator to be defined?
The theory as I know it
Let $\mathcal{H}$ be a Hilbert space and $(A, D(A))$ a self-adjoint operator acting on it. The Spectral Theorem (cfr. Reed & Simon Methods of modern mathematical physics, ...
5
votes
1answer
874 views
What's the connection between the Laplace transform and the Fourier transform?
Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
1
vote
3answers
204 views
references for the spectral theorem
Recently, I am thinking about the question in spectral theory. And I finally found that I need help with the properties of unitary operator. Its a consequence of the spectral theorem for the normal ...
5
votes
1answer
435 views
spectrum of the “discrete Laplacian operator”
In numerical analysis, the discrete Laplacian operator $\triangle$ on $\ell^2({\bf Z})$ can be written in terms of the shift operator
$\triangle=S+S^*-2I$
where $S$ is the right shift operator. ...
1
vote
2answers
315 views
Basic Spectral Theory Problem: Finding the Point/Continuous Spectrum of an Operator
I have the following problem:
Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for ...
