# Questions tagged [spectral-theory]

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

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### Question about proof that normal operators have invariant subspace

In Normal $T\in B(H)$ has a nontrivial invariant subspace, Haskell Curry said (for case (ii)) to pick two open, disjoint subsets of $\sigma(T)$ where $|\sigma(T)|\geq 2$. I don't think that this is ...
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### Prove that the limit of an operator sequence is infinite [closed]

Let $X$ be a Banach space, $G$:={$A$ is an operator : $A$ and $A^{-1}$ $\in$ $L(X)$}.Suppose $T$ is a boundary point of $G$, and ${T_n}$ is an operator sequence which is in $G$ and $L(X)$.$T_n$ tends ...
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### essential range of function,range and spectrum

hi here $\sigma$ is spectrum,if $(X,\mu)$ is topological measure owed with borel measure and $$r_{ess}(f) = \{w \in \mathbb{R}_+ : \mu(f^{-1}(B(w, \epsilon))) > 0\}.$$ i have proved that if f is ...
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### Self adjoint operator $A$ satisfies $\|A(A-i)^{-1}\|\leq 1$?

Let $H$ be a seperable Hilbert space and $A=A^*:D\subset H \rightarrow H$ a possibly unbounded linear densly defined selfadjoint operator. I try to understand a proof in a script, where a crusial step ...
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### Theorem (Rellich) - Perturbation Theory

I got stuck with part of a proof of: The steps are all clear to me, until it is said: "Therefore $f_w$ is the eigenvector of $A+wB$ associated with the eigenvalue lying within for all small ...
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### What properties of an operator can be deduced by looking at its approximate point spectrum?

May I ask something to the community: If $A: H \rightarrow H$ is a normal, compact operator on a complex Hilbert space H, then $\sigma(A)=\sigma_p(A)\cup \lbrace 0\rbrace$, since if $z-A$ $(z\neq 0)$ ...
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### Suficient condition for an unital $C^*$-algebra homomorphism to be surjective on postive elements.

I am trying to find sufficient conditions for an unital $C^*$-algebra homomorphism $\alpha:A \rightarrow B$ to be surjective on positive elements, that is, $\alpha(A_+)=\alpha(B_+)$. For example, if ...
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### Nilpotency of $(A-z)P_z$ imply $z$ is an Eigenvalue?

Let $z$ be an isolated point in the spectrum of some possibly unbounded operator $A:D\subset X \rightarrow X$ on a Banach space $X$. Where the spectrum $\sigma(A)$ without $z$ is closed. Let $P_z$ be ...
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### Convergence of self-adjoint operators with converging spectra

Let $A=\int_{\sigma(A)} \lambda dP(\lambda)$ be an unbounded self-adjoint operator on a Hilbert space $H$ with dense domain $D\subset H$. Let $A_n=\int_{\sigma(A)\cap[-n,n]} \lambda dP(\lambda)$ (see ...
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### How to make sense of the spectrum of an unbounded operator

I'm trying to work my way up through various definitions in order to understand the formulation of the spectral theorem for unbounded operators, in which figure projection valued measures. I'm having ...
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### Continuous function of a projection operator is itself?

Let $\{\psi_{n}\}_{n\in \mathbb{N}}$ a sequence of unit elements of a given Hilbert space $\mathscr{H}$. These are not necessarily orthogonal. For each of these vectors, define the projection operator ...
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### Infimum of numerical range = infimum of spectrum?

Let $\mathcal{H}$ be a Hilbert space and $A: D(A) \to \mathcal{H}$ a densely defined self-adjoint operator. Its numerical range is defined to be the set $\{\langle x, Ax\rangle: \|x\| =1\}$. My ...
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### Discrete spectrum of laplacian $\sigma_{disc}(\Delta)$ on $H^2(\mathbb{R}^n)$?

Let $\Delta: H^2(\mathbb{R}^n)\subseteq L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$ be the Laplace operator in the weak sense. What is the discrete spectrum of $\Delta$? the discrete spectrum ...
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### Intuition behind the spectral theorem in infinite dimensions

I would like to verify my intuition behind the spectral theorem in infinite dimensions. For the moment I am putting bounded/unbounded and domain issues aside. In finite dimensions the spectral theorem ...
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### Spectral theorem for diagonal matrix in different inner product spaces

I learned a special case of the spectral theorem for finite dimensional inner product space. As I understand it states that a real matrix is orthogonally diagonalizable with real eigenvalues iff it ...
### Is $\|u\|_\infty\leq C\|u\|_{L^2}$ true for $u \in H^2(\mathbb{R}^{n})$ for $n>3$?
The Sobolev Lemma says if $u \in H^m(\mathbb{R}^n)$ $(m\in \mathbb{N})$, $k\in \mathbb{N}_0$ such that $k<m-n/2$ then $u\in C^k(\mathbb{R}^n)$ and for $|\alpha|\leq k$ \sup\limits_x |\partial^\...