0
votes
1answer
16 views

Orthonormal basis $L^2(a,a+2\pi)$

Let $$\mathcal{B}=\left \{\frac{1}{\sqrt{2\pi}},\frac{\cos x}{\sqrt{\pi}},\frac{\sin x}{\sqrt{\pi}},\frac{\cos 2x}{\sqrt{\pi}},\frac{\sin 2x}{\sqrt{\pi}},\dots\right \}$$. This is an orthonormal basis ...
4
votes
1answer
64 views

Composition of two orthogonal projections

Let $V$ be a finite dimensional Euclidean space and let $W_1,W_2$ be two subspaces of $V$. Let $P_1,P_2$ denote the projections onto $W_1,W_2$ respectively. Is it true that the composition $P_1\circ ...
0
votes
0answers
122 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
6
votes
1answer
94 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
0
votes
2answers
68 views

Minimum of $f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$

Show that $$f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$$ is defined on $\Bbb{R}^2$, it has a minimum and find for which couple $(x, y)$ the minimum is reached. The first point is okay, ...
8
votes
2answers
121 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
0
votes
2answers
26 views

Proof: adjoint map of projection is a projection and …

Let $V$ be a pre hilbert space and $\pi \in \mathrm{End}(V)$. Show: the adjoint map $\pi^+$ of a projection (meaning: $\pi^2 = \pi$) is a projection itself. Show then: a projection $\pi$ is ...
2
votes
4answers
88 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
1
vote
0answers
16 views

The relationship between CPTP maps and quadratic forms

Let $H$ be a finite-dimensional Hilbert space (so there is a canonical isomorphism $H\cong H^*$). For a Hilbert space $H$ define $B(H)$ to be the space of linear operators on $H$; we have $B(H)\cong ...
1
vote
1answer
25 views

Is the image of a closed subspace under a bounded linear operator closed?

This seems obvious, but I can't get the proof straight, and I made up the statement myself, so I'm not sure if it's true in the stated generality. Given a bounded linear operator $T$ in Hilbert space, ...
1
vote
0answers
38 views

Hilbert space without the projection theorem

One succinct statement of the projection theorem in Hilbert space is $A+A^\bot=\scr H$, where $A\in\scr C$, the set of closed subspaces of $\scr H$. (We will also denote the set of all subspaces by ...
2
votes
1answer
38 views

Operator norm of orthogonal projection

I was assigned the following homework problem: "Let $P:\mathcal{H} \to \mathcal{H}$ be bounded and linear. Assume it satisfies $P^2 = P$ and $P^\star = P$. Show $\|P\| \le 1$." This isn't too hard ...
0
votes
0answers
28 views

The subspace sum of closed subspaces is closed [duplicate]

Given an arbitrary Hilbert space $\scr H$ and closed subspaces $A,B\subseteq\scr H$ with trivial intersection, is it true that $A+B=\{x+y:x\in A,y\in B\}$ is closed? So far, I have the following: Let ...
1
vote
1answer
54 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty ...
1
vote
1answer
32 views

Show that a Hilbert space with two inner products has a basis that is orthogonal with respect to both inner products

Let $\mathcal{H}$ be a complex, $n$-dimensional Hilbert space with two inner products $\langle \cdot, \cdot \rangle_1$, $\langle \cdot, \cdot \rangle_2$. Show that there exists a basis $ X = x_1, ...
0
votes
0answers
20 views

A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$ for all x

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
1
vote
1answer
48 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
0
votes
1answer
9 views

Projection with modulus less than one

Let $X$ be an Hilbert Space, $X=Y\bigoplus Z$ where $Y$, $Z$ are both closed subspaces. Let $P:X \rightarrow X$ $P(y+z)= y$ be the canonical projection, then $||P|| \leq 1 \implies Y=Z^{\bot}$ ...
1
vote
2answers
42 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
2
votes
1answer
23 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
0
votes
2answers
65 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
0
votes
0answers
38 views

Generalisation of Gramian determinant

i'm wondering about those facts of basic linear algebra: if you have $n$ vectors $x_1,...,x_n \in \mathbb{R}^n$, you can easily test their linear dependance by computing their Gramian Matrix $M$ whose ...
0
votes
3answers
87 views

polar decomposition proof

Let $H$ be a hilbert space and $T$ a bounded linear operator on $H$. I'm trying to prove that there is a partial isometry $V$ on the closure of $Im(|T|)$ such that $T=V|T|$ and $|T|=V^*T$, where ...
2
votes
2answers
36 views

Showing $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$

My book asserts that for fixed $w$ where $w\neq 0$ that $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$ My book has a general corralary that $v\to P(v)$ is a bounded linear transformation and ...
1
vote
2answers
66 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
2
votes
1answer
102 views

$\langle Tx,x \rangle=0$ proof

If $T$ is a bounded operator on a hilbert space $H$ and $\langle Tx,x \rangle=0$ for all $x$ in $H$, then $T=0$. I'm considering what we can conclude if $\langle Tx,x \rangle=0$ for all $x$ in some ...
0
votes
2answers
55 views

Rudin Polar decomposition

On page 332 theorem 12.35b) of Rudin functional analysis is show that if T is normal then it has a polar decomposition $T=UP$. Does he mean that $P=|T|$? He's a bit ambiguous as to how he defines ...
0
votes
2answers
81 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...
1
vote
1answer
35 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
1
vote
3answers
129 views

Borel functional calculus

For a normal operator T, we have a resolution of the identity $\int_{{\sigma}(T)} {\lambda}\,dE=T$. If $T$ is in addition compact , we have that $\sum_{n=1}^{{\infty}}{\lambda}_{n}\langle ...
4
votes
1answer
83 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
0
votes
2answers
84 views

Cauchy-Schwarz Inequality by sum of squares.

i'm trying to solve: Prove the Cauchy-Schwarz inequality by writing $||x||^2||y||^2− |⟨x,y⟩|^2$ as a sum of squares. I'm fairly well versed in Cauchy Schwarz and know several proofs but I'm confused ...
2
votes
1answer
51 views

Polar decomposition corollary

Let $T$ be a compact operator on an infinite dimensional Hilbert space. Let $|T|=(T^*T)^{0.5}$. By the polar decomposition theorem there is a partial isometry $S$ of the closure of Im$(|T|)$ such that ...
4
votes
0answers
99 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
2
votes
1answer
73 views

Positive compact operator has unique square root.

Let H be a hilbert space and T be a compact positive operator so that by the spectral decomposition theorem, $T=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ where the $e_{n}$ are the ...
0
votes
2answers
69 views

Checking whether an operator is self-adjoint. Problem with domain of an operator.

I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The ...
0
votes
0answers
53 views

Hermitian and self-adjoint operator.

I was trying to understand why this operator is hermitian (I see that) but not self-adjoint: We have the operator $P:\phi(\cdot )\mapsto -i\phi '(\cdot )$, dfined in $C_0^\infty (I)$ (infinitely ...
2
votes
2answers
124 views

Questions about the proof of the Riesz representation theorem

Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map. I'm trying to prove that there exists only one $z_0 \in H$ such that: $ \forall_{x \in H} : f(x)=\langle x,z_0\rangle$ ...
3
votes
3answers
97 views

Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
0
votes
1answer
26 views

Hilbert space continuous linear map, one dimensional subspace

Could you help me with the following exercise? Let $H$ be a Hilbert space, $\alpha : H \rightarrow \mathbb{C}$ a linear continuous mapping, $\alpha \neq 0$. Prove that the orthogonal complement ...
4
votes
2answers
158 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
1
vote
3answers
73 views

Prove a space is Hilbert [duplicate]

I got stucked in this problem and get no clue to solve this. Can any one please help me? Thanks Suppose $X$ is an inner product space. If for every bounded linear function $f$, there exists $z \in ...
2
votes
1answer
419 views

Orthonormal Basis for Hilbert Spaces

The following is the definition of orthonormal base that I am using: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an ...
1
vote
0answers
48 views

How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := ...
1
vote
1answer
202 views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
1
vote
0answers
59 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
1
vote
1answer
87 views

What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
3
votes
1answer
102 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
0
votes
3answers
102 views

Showing that inner product of two vectors is the limit of the inner products

How can you show that $$(a,b) = (\sum_{i=1}^\infty a_i \phi_i,\sum_{i=1}^\infty b_i \phi_i) = \lim_{N\to \infty}(\sum_{i=1}^N a_i \phi_i,\sum_{i=1}^N b_i \phi_i)$$ where $\phi_i$ is an orthonormal ...
2
votes
1answer
51 views

a question on decreasing sequence of subspaces

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...