1
vote
1answer
22 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 $||P|| \leq 1 \implies Y=Z^{\bot}$ ...
0
votes
2answers
32 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
12 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
51 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
24 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 ...
1
vote
3answers
51 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
35 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
58 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
80 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
48 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
51 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
30 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
100 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
55 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
58 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
43 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
90 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
48 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
56 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
29 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
115 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
75 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
22 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
103 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
51 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
273 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
42 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 := ...
0
votes
0answers
32 views

Please help understand a particular proof of basic theorem

I read the following proof that in a vector space $V$ of dimension $n$, a set of orthonormal vectors $\{\phi_1, ..., \phi_m\}$, with $m<n$, is not complete : Among the linear combinations ...
4
votes
1answer
120 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
58 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
71 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
90 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
80 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
47 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 ...
1
vote
2answers
70 views

Basis means determinant of matrix of inner products is non-zero

Let $x_i$ be a basis of Hilbert space $X$ (NOT necessarily orthogonal) How do I show that $\text{det}((x_i,x_j)_H)_{ij} \neq 0$ for $i,j=1,...,n$? I see this fact used in Galerkin approximation ...
2
votes
2answers
91 views

Find the fallacy in using the Cauchy–Schwarz inequality

Let $\int_{a}^{b}\frac{f(x)}{x}dx=k$, wherein $f(x),a,b,k$ are positive. According to the Cauchy–Schwarz inequality: $\int_{a}^{b}xf(x)dx=\int_{a}^{b}x^{2}\frac{f(x)}{x}dx\leq \left ( ...
4
votes
2answers
136 views

Hermitian matrices and great circles

I am considering parameterised curves in an $n$-dimensional complex vector space, given by the solution to the discrete Schrödinger equation: $$ |\psi\rangle(t) = e^{-iHt}|\psi_0\rangle, $$ Where $H$ ...
2
votes
1answer
86 views

Calculating the norm of an infinite vector

I'm reading "Introduction to Hilbert Spaces" by N. Young. Right in the first chapter, after introducing inner products and norms in general linear spaces, it asks to show that the norm of the vector: ...
1
vote
2answers
76 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
1
vote
1answer
134 views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
15
votes
4answers
338 views

How to interpret the adjoint?

Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$. I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I ...
5
votes
0answers
98 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
2
votes
1answer
109 views

if $E^2=E$, then $\text{Im}\;E\subset\left(\ker E+(\ker E)^\perp\right)$?

Notation: $V$ is a infinite-dimensional inner product space; $\langle\cdot,\cdot\rangle$ is the inner product of $V$; $E:V\rightarrow V$ is a linear map; $\text{Im}=\{E(v):v\in V\}$; $\ker E=\{z\in ...
3
votes
1answer
213 views

A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
2
votes
0answers
49 views

Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
2
votes
2answers
58 views

little question about linear operators

Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
1
vote
1answer
48 views

some inclusions regarding linear operators

Let $H$ be a Hilbert Space and $T:H\rightarrow H$ a linear operator. Let $T^*$ be the adjoint operator of $T$ and let $\operatorname{Cl}(X)$ be the topological closure of the set X and $X^{\perp}$ ...
1
vote
0answers
55 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
2
votes
1answer
78 views

inner product space and injective -surjective

Let $V$ and $W$ be two finite-dimensional inner product spaces over the same field and let $T\in \mathcal{L}(V,W)\ $ be a linear transformation. Show that $T$ is injective if $T^*$ is surjective.