For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional?

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional, for any $\lambda$ in A's spectrum? P.S: What I know now is that the spectrum of A is discrete.
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15 views

Calculate the trace of $T_nL$ where $L\in L(H)$, $T\in L(H,L(H))$ and $T_n:=\langle T,e_n\rangle_H$ for some ONB $(e_n)_n$ of a Hilbert space $H$

Let$^1$ $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
4
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2answers
512 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As ...
9
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1answer
292 views
+100

Approximate spectral decomposition

See attempt below I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a Hermitian ...
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0answers
18 views

precompact operators in a Hilbert space [functional analysis]

I've linked to a Theorem (from H&N's Applied Functional Analysis) whose proof I'm trying to understand (I asked a question about the previous chunk of the proof yesterday). The theorem is ...
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1answer
22 views

Operator groups

In $H := L^2(\mathbb{R}, \lambda)$ Hilbert-space, the following two, one-variable operator groups are given: $$(U_s f)(x):=f(x-s)$$ $$(V_s f)(x):=e^{is x} f(x)$$ $f \in H, s \in \mathbb{R}$. a, ...
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0answers
37 views

What would be an arrow in category of Hilbert space?

Let $H,K$ be Hilbert spaces. Let $S$ be a Hilbert basis for $H$. (Which means that $S$ is orthogonal and the span of $S$ is dense in $H$.) For each arrow $S\rightarrow K$, there exists a unique ...
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34 views

Operator continuity on Hilbert space

Let $A: H \to H$ be a linear operator on Hilbert space $H$, and let $\{\alpha_n\}_{n = 1}^{\infty} \subset \mathbb{R}$ converges to nonzero number. Prove that if the series $\sum_{n = 1}^{\infty} ...
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0answers
7 views

Example for injective and surjective bounded and unbounded operator

I am looking for some bounded and unbounded densely defined operators on a real Hilbert space $H$, let say $A:D(A)(\subset H)\to H$, that are one-to-one but they are not onto. I am wondering whether ...
3
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22 views

Showing a C* Algebra contains a compact operator

In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem: Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset ...
3
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0answers
30 views

Proving/ Disproving that a set is compact in $l^2$

How can I prove or disprove that the following set in the real sequence space $l^2$ ( equipped with the norm $||(X_1,X_2,...)||_2 = \sqrt {\sum_{i=1}^{\infty} X_i^2}$ ) , is compact? $$ A = ( ...
3
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3answers
27 views

multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...
5
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2answers
48 views

For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
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0answers
39 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
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2answers
703 views

Closed unit ball in infinite dimensional normed linear space

I have to prove that in any infinite dimension normed linear space we have that the closed unit ball is not compact. I know that I have to construct a sequence such that $||x_n||=1$ and ...
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0answers
28 views

Adjoint of differential operator

I would like to find the adjoint of the operator $T_a$ ($a\in \mathbb{C}$) on $ \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$ with $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ ...
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24 views

Is $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$?

I'm trying to learn about tensor products of Hilbert spaces and started to wonder if $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$? If $(e_n)$ denotes the standard basis, in the first case, it ...
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1answer
8 views

Can a span of an orthonormal subset be embedded into $l^2$?

Rudin - RCA p.85 Let $H$ be a Hilbert space and $\beta$ be an orthonormal subset of $H$. Let $T:H\rightarrow l^2(\beta)$ be the continuous linear transformation such that $T(x)(v)=(x , v)$ for $x\in ...
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3answers
46 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
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1answer
24 views

Significance of closedness of a subspace when writing a Hilbert space as a direct sum

I read that if $U$ is a closed subspace of a Hilbert space $H$ then we can write $H$ as $H = U \oplus U^\bot$ (the direct sum). What is not clear to me is why $U$ is required to be closed. I thought ...
5
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0answers
72 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
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1answer
19 views

Limit of sesquilinear forms is a sesquilinear form

Suppose $P_n$ is a monotone sequences of orthogonal projections in a complex Hilbert space $\mathcal{H}$. I want to show that the limit of the sesquilinear forms defined by: $\Gamma_n(x,y)=(x,P_n y)$ ...
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30 views

Hilbert Space, showing a sequence in Cauchy

Suppose $X$ is a Hilbert space, $M\subset X$ is a closed subspace and $y\notin M$. Let $d = \inf\{ \|x-y\|:x\in M\}$ show that if $\{x_n\}_{1}^{\infty}$ and $\lim_{n\rightarrow \infty}\|x_n - y\| = ...
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1answer
471 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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3answers
260 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
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1answer
21 views

Operator Sum: Selfadjoint

Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}A\to\mathcal{H}:\quad A=A^{**}$$ Does it follow that: $$S:=\overline{A+A^*}:\quad S=S^*$$ (Rigorous proof?) Densely ...
2
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2answers
14 views

Monotone sequence of orthogonal projections on a complex Hilbert space

Suppose $P_n$ is a monotone sequence of orthogonal projections on a complex Hilbert space $\mathcal{H}$, i.e. $V_n= Im(P_n)$ is a decreasing or increasing sequence of subspaces and $P_n^\star=P_n$ and ...
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0answers
14 views

Characterization of noncommutative $L^2$-spaces as ordered vector spaces

If $M$ is a von Neumann algebra and $\tau\colon M_+\to[0,\infty]$ is a normal, semi-finite, faithful trace, the associated GNS Hilbert space is the completion of $\{x\in M\mid \tau(x^\ast ...
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1answer
19 views

Orthonormal basis for Hilbert space

Usually, an orthonormal basis for Hilbert space means an orthonormal subset which unconditionally spans the whole space. However, I'm curious whether there exists an orthonormal basis for every ...
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0answers
14 views

Separable Hilbert space convergence problem

Suppose that $\{x_j \}_{1}^{\infty}$ is a sequence of separable Hilbert space $X$ and that $\|x_j\| \leq 1$ for all $j$. Show that there is a subsequence $\{x_{j_k} \}_{k=1}^{\infty}$ such that for ...
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0answers
21 views

Does $S^{\perp\perp}=\overline{S}$?

I'm trying to prove that $S^{\perp\perp} = \overline{S}$, where S is a subspace of a Hilbert Space, where $S^{\perp\perp} = \{f: <f,g> = 0, \forall g\in S^\perp\}$ and $\overline{S}$ is the ...
2
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1answer
26 views

Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_n$, what's the second derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle ...
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3answers
28 views

If $S$ is a subset of a Hilbert space $H$ then $S^\perp$ is closed?

Is this a true statement? (I found it as a theorem in a paper) If $S$ is a subset of a Hilbert space $H$ then $S^\perp$ is closed. If it were true then $(S^\perp)^\perp$ would be closed, that is ...
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1answer
39 views

Sequence in hilbert space, mutually orthogonal vectors

Let $y_1,y_2,\cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1,\cdots,y_n\}$. Assume that $||y_{n+1}||\leq ||y-y_{n+1}||$ for all $y\in V_n$ for $n=1,2,3,\cdots$. Show ...
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1answer
18 views

Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_{n\in\mathbb N}$, what's the derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle ...
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0answers
12 views

Some property of a sequence in Hilbert space [duplicate]

Let $y_1, y_2, \cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, \cdots, y_n\}.$ Assume that $||y_{n+1}|| \leq || y -y_{n+1}||$ for each $y \in V_n$ for $n = 1, 2, ...
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1answer
254 views

Operator self-adjoint

I have this paragraph : "Let M be a Hilbert-Riemannian manifold. $f \in C^2(M,R), p \in K$ is called a nondegenerate critical point, if $d^2 f (p)$ has a bounded inverse. Since $A = d^2 f (p)$ is a ...
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0answers
10 views

Gaussian Hilbert spaces indexed by a Hilbert space

Let $H$ a real Hilbert space. Then, there is a real Gaussian Hilbert space $G$ indexed by $H$. I know this result is a consequence of Kolmogorov Extension Theorem, but I have not idea of how ...
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32 views

The space of sequences which are eventually zero in $l^2$ is not a Hilbert space.

Define $V$ to be the space of sequences which are eventually zero, i.e. $$V=\bigcup_{N=1}^{\infty}\{(x_n)_{n=1}^{\infty}\in l^2: x_n=0 \; \text{for}\; n\ge N\}.$$ Is $V$ a Hilbert space with ...
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1answer
26 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
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1answer
30 views

Finite Dimensional Hilbert Space

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x ...
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1answer
28 views

Definition of inner product space

In the definition, we defined linearity in the first argument, Hermitian symmetry. And these two imply anti-linearity in second argument. Is it equivalent, if I cancel the Hermitian symmetry and only ...
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1answer
25 views

If $(U,〈\;⋅\;,\;⋅\;〉),H$ are Hilbert spaces, $W\in U$, $Y\in H$, $Z\in L(U, H)$ and $f\in L(H,L(H,\mathbb R))$, then $〈Y,fZW〉=〈ZW,fY〉$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be Hilbert spaces $W\in U$, $Y\in H$ and $Z\in\mathfrak L(U, H)$$^1$ $f\in\mathfrak L\left(H,\mathfrak L\left(H,\mathbb R\right)\right)$ How ...
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2answers
26 views

Let $C$ be a convex closed nonempty subset of a Hilbert space $H$. Show that there is a unique element in $C$ with the minimum norm.

Let $H$ be a Hilbert space and $C \subset H $ be a convex, closed and nonempty subset of $H$. Prove that there exists a unique $x_0\in C$ with minimum norm among the elements of $C$. I don't know ...
0
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1answer
25 views

Lipschitz Homeomorphism and $\mathcal{l_p}$ spaces

While solving a homework that asked me to find two metric spaces that are not lipschitz homeomorphic it crossed my mind the question (assuming to have in $\mathcal{l^n}$ the metric ...
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1answer
26 views

$L^2$ convergence of Taylor series of a holomorphic function

I am reading Otto Forster's book "Lecture on Riemann surfaces" and on pages 109-110, he introduces the space $L^2(D,\mathcal{O})$ of holomorphic square-integrable functions $f:D\to \mathbb{C}$ (where ...
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1answer
14 views

positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...
2
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1answer
36 views

If $H$ is a Hilbert space, are we able to identify the derivative ${\rm D}f(x)$ at some $x\in H$ of a differentiable $f\in H'$ with an element of $H$?

I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not: Let ...
0
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2answers
18 views

summing inner product of orthonomal basis

I need some help with some very basic linear algebra when doing calculations in inner product space. Here is a line I got lost when reading a book... \begin{align*} ...