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

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8
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2answers
177 views

Prove of inequality under a Hilbert space.

Let $x\neq y$ when $x,y\in H$ and H is a Hilbert space which satisfy $\|x\|=\|y\|=r$. Show that $\|\frac{x+y}{2}\|<r$. Actually in my question r=1 but as far as i could understand there is a way ...
1
vote
1answer
29 views

Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Let $\Omega\subset \mathbb{R}^d$, with $d\in \{1,2,3\}$ be an open bounded, simply connected domain. Define $H_0^1$ as the subspace of $H^1$ whose member functions have vanishing trace on the ...
0
votes
1answer
33 views

Properties of Injective Operator on Hilbert Space

I am new to functional analysis and have the following issue: Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional ...
0
votes
1answer
25 views

Space filling curves: initial definitions

I am confused on the definition of curve and space filling curve in Chapter 1 of the book by Sagan. I think my confusion comes from notation. Let $\mathcal{I}:=[0,1]$, $\mathcal{Q}:=[0,1]^2$ and $J_n$ ...
1
vote
1answer
48 views

An expression for the Hilbert-Schmidt inner product

Suppose that $k:[0,1]\times[0,1]\to\mathbb C$ is a Hilbert-Schmidt kernel, i.e. $$ \int_0^1\int_0^1|k(x,y)|^2\mathrm dx\mathrm dy<\infty. $$ The associated Hilbert-Schmidt integral operator $K:L^2([...
2
votes
0answers
26 views

Von Neumann algebraic Quantum Object is direct sum of type I factors

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra ...
0
votes
1answer
15 views

Product of Lebesgue measure on Hilbert cube doesn't satisfy doubling condition?

The Hilbert cube $H$, is the infinite dimensional product $[0,1]\times [0,\frac12]\times...$ Let $\mu$ be product of Lebesgue measures $\mathcal{L}^1 \times \mathcal{L}^1\times...$, I heard that the ...
1
vote
1answer
23 views

Functionals taking real values

Suppose $f$ is a bounded functional on a separable Hilbert space. Can we always find an orthonormal basis such that $f$ takes real values on that basis?
0
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0answers
17 views

How to prove that $ e_{\lambda}$ can be written in the following form?

Let $e_{\lambda}$ be the spectral density associated to the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex Hilbert space $(H,\left<., .\right>)$. Haw to prove that ...
1
vote
1answer
27 views

norm from inner product

I have a question in a Hilbert Spaces course as follows: Let $X=(x_1, x_2)$ be vector in a vector space of all ordered pairs of complex numbers X. Can we obtain the norm defined on X by: $\|X\|=|x_1|...
0
votes
1answer
34 views

unitary operator between two Hilbert subspaces

$H$ is a Hilbert space. $P, Q$ are projections. For every $x\in P(H)$, we have decomposition $x = Qx +Q^\perp x$. Then, can we find a unitary operator from the space generated by all $Qx$, $x\in P(H)$...
2
votes
2answers
41 views

Norms with complex numbers over Hilbert Spaces

Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar. Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. I want to choose a such that $\bar{a}(v,w)$ ...
2
votes
1answer
46 views

projections in von Neuman algebra

Consider a semifinite von Neumann algebra $\mathcal{M}$ with a semifinite faithful normal trace $\tau$. If $Q, P$ are projections in $\mathcal{M}$ with $\tau(Q)< \tau(P)$, then does $\tau(P\wedge Q^...
1
vote
1answer
23 views

Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$ \operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}. $$ (it is a cone in ...
0
votes
0answers
10 views

Bounded input-Bounded output stability for countable system of ODES.

Let $X$ be a countably infinite dimensional Hilbert space. Let $f\colon X\to X$ be a compact, linear, symmetric positive definite map. Define an ODE as $u_t = -f(u-y)$ and $u(0) = 0$, where $y\...
3
votes
1answer
73 views

How can we prove that the space of trace class operators on a Hilbert space $H$ is the closure of $H\otimes H$ with respect to the trace norm?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space over $\mathbb R$ $\mathfrak L^1(H)$ be the space of trace class operators on $H$ and $$\operatorname{tr}L:=\sum_{n\in\mathbb ...
0
votes
0answers
37 views

Is there a name for these inequalities? Where can I look them up?

Consider the operators $A,B,C$ on Hilbert space $\mathcal H$: Show that: $$ \left \vert \left \vert AB \right \vert \right \vert \le \left \vert \left \vert A\right \vert \right \vert \left \vert \...
7
votes
1answer
83 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
1
vote
1answer
18 views

Almost negative definite matrices and norm-distance matrices

An "almost negative definite" matrix $A$ satisfies the property $$ v^te = 0\implies v^tAv\le 0 $$ where $e=(1,1,\dots,1)$. We know that if $A$ is a simmetric zero-diagonal (hollow) matrix, then $A$ ...
0
votes
0answers
28 views

Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
0
votes
1answer
31 views

Showing that the intersection of two closed linear subspaces is the trivial subspace.

I'd appreciate if someone can provide the best way to deal with this problem. Let $\{\alpha_n\}$ be an orthonormal sequence for a Hilbert space H and let $\{\beta_n\}$ be an orthonormal sequence such ...
1
vote
2answers
49 views

Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
2
votes
1answer
30 views

Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
2
votes
1answer
40 views

Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
3
votes
1answer
27 views

If operators $A, B, A+B$ are all closable, show that $\overline{A + B} \supseteq \overline{A} + \overline{B}$.

Let $A, B$ be closable, unbounded, linear operators on a Hilbert space $H$. Suppose further that the operator $A + B$, defined on the intersection of domains $D(A) \cap D(B)$, is also closable. I ...
0
votes
1answer
23 views

the orthogonal complement intersection of sets

Let $\{C_\gamma\}$ be a net of subsets of a Hilbert space(or some other spaces). Do we have $$(\wedge_\gamma C_\gamma )^\perp = \vee_\gamma C_\gamma ^\perp?$$ It is known that it works when $\{C_\...
60
votes
7answers
5k views

What is “Bra” and “Ket” notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my ...
3
votes
0answers
37 views

Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete

Consider an operator $H=-\Delta +U(x)$ on $L^2(\mathbb R)$ for a function $U(x): \mathbb R \to \mathbb R$ that tends to $+\infty$ as $|x|$ grows. These kinds of operators appear all over non-...
0
votes
1answer
18 views

Show a function is Sobolev

Let $T_{h}$ be a subdivision of a domain $\Omega \subset \mathbb{R}^d$ into elements $K$ with boundary $\delta K$so that the Gauss divergence theorem holds. If for a function $f$ it holds that $f \...
2
votes
1answer
30 views

Tomita Theory: Involution

Given a Hilbert space $\mathcal{H}$. Consider a von Neumann algebra: $$M\subseteq\mathcal{B}(\mathcal{H}):\quad M=M''$$ Suppose a cyclic vector: $$\Omega\in\mathcal{H}:\quad\overline{\mathcal{M}\...
0
votes
2answers
46 views

Function $f$ such that $f''\in L^2(\mathbb{R})$

Let us assume that $f\in L^2(\mathbb{R})$ and $f''\in L^2(\mathbb{R})$ ($f'$ - first derivative,$f''$ - second derivative), i.e. $f$ is square-integrable, $f$ is differentiable, its first deriviative ...
1
vote
0answers
18 views

Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde Q(x):=...
0
votes
0answers
17 views

Extending a unitary isomorphism on a Hilbert space

Let $H$ be a Hilbert sapce and $M$ a dense subspace of $H$. Prove that any unitary isomorphism on $M$ can be uniquely extended to a unitary isomorphism on $H$. So here's what I tried: Let $T: M \...
1
vote
0answers
18 views

Characterizing when kernels in Reproducing Kernel Hilbert Space (RKHS) are linearly independent

In my studies of RKHS i.e. Reproducing Kernel Hilbert Spaces stating the following Let $ \mathbb{H} $ be a RKHS on a set X. We are asked to characterize when the following set of kernels $ \{ k_{...
0
votes
1answer
18 views

some detail calculation on the proof of equivalence of norms

We say that two norm $\|x\|_1$ and $\|x\|_2$ on a vector space $X$ are said to be equivalent if there exists $K>0$ and $M>0$ such that $$ K\|x\|_1\le \|x\|_2\le M\|x\|_1 $$ Prove that on a ...
3
votes
3answers
76 views

Why are function spaces generally infinite dimensional

The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ...
0
votes
1answer
49 views

Approximate point spectrum of a normal operator

how can I show the following theorem? Let $H$ a Hilbert space and $T:H \to H$ a linear, continuos and normal operator. Then for every $\lambda \in \sigma(T)$ there exists a sequence $(x_n)_{n \in \...
1
vote
1answer
25 views

Spectral projections, additivity

Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is $$ E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} (1,\...
1
vote
0answers
20 views

Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$

The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...
3
votes
0answers
34 views

Show that for the triples $V \subset H \subset V^{*}$, the following are true

Let $H$ be a Hilbert space equipped with scalar product $(,)$ and the corresponding norm $|\cdot|$. Let $V \subset H$ be a linear subspace that is dense in $H$. Assume that $V$ is a Banach space for $\...
1
vote
1answer
27 views

Explicit example of two non-isomorphic Hilbert spaces with the same algebraic dimension [duplicate]

I´m wondering if there exist a vector space A and inner products: $\langle\cdot{,}\cdot\rangle_1$ and $\langle\cdot{,}\cdot\rangle_2$, such that: $\big( A,\langle\cdot{,}\cdot\rangle_1 \big)$ and $\...
1
vote
0answers
32 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 $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
0
votes
0answers
20 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 ...
1
vote
1answer
23 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, ...
2
votes
0answers
41 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 ...
4
votes
0answers
52 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} \...
0
votes
1answer
18 views

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.
0
votes
0answers
13 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
votes
0answers
32 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 B(\mathbb{H}...
3
votes
0answers
31 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 = ( (X_1,...