0
votes
1answer
18 views

Is tensor product commutative on orthonormal basis?

In general the tensor product $\varphi\otimes\psi$ is not commutative, but I was thinking that if I have tensor product on two orthonormal bases of Hilbert spaces are they commutative i.e is ...
0
votes
1answer
16 views

Identification of tensor product spaces

Let $P_1, P_2$ be (probability) measures, $\Omega_1, \Omega_2 \subset \mathbb{R}^n$ . Prove that $L_{P_1 \otimes P_2}^2(\Omega_1 \times \Omega_2)$ and $L_{P_1}^2(\Omega_1) \otimes L_{P_2}^2(\Omega_2)$ ...
0
votes
0answers
7 views

Universal property of tensor products of real Hilbert spaces

I have the following exercise where I could need some hints: Let H1, H2 be real Hilbert spaces. Prove that there is a weak Hilbert-Schmidt mapping $$ p: H_1 \times H_2 \rightarrow H_1 \otimes H_2 ...
1
vote
1answer
103 views

Gelfand triple for tensor product of Hilbert spaces

Is there any dense embeding $\to$ that makes $H^1_0(D) \otimes L^2(\Gamma) \to L^2(D) \otimes L^2(\Gamma) \to (H^1_0(D) \otimes L^2(\Gamma))^{*}$ a Gelfand tripe? In fact we may only answere to the ...
0
votes
0answers
57 views

Gelfand triple in Tensor product spaces

Let's define $A^k={\rm span} \{\phi_1,\dots,\phi_k\}$ and $B^k={\rm span} \{\psi_1,\dots,\psi_k\}$ where $\phi \in H^1(X)$ and $\psi \in L^2(X)$ simple functions and $X$ is closed subset of ...
2
votes
1answer
48 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
3
votes
0answers
59 views

What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
0
votes
0answers
94 views

Forming the tensor product of a `real' vector space with a 'complex' vector space.

I have a question that I am hoping someone could clarify for me. Context: Consider the algebra $A = (B,\circ)$, given by: \begin{align} B = \{ \begin{pmatrix} a & f\\ \overline{f} & ...
3
votes
1answer
90 views

The definition of addition on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and ...
4
votes
2answers
438 views

Tensor product of operators

We know that if $T_1$ is a linear bounded operator on a Hilbert space $H_1$ and $T_2$ is a linear bounded operator on a Hilbert space $H_2$ there exists a unique linear bounded operator $T$ on $H_1 ...
1
vote
0answers
94 views

Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?

Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$ H = H_1 \otimes \cdots \otimes H_n, $$ and let $\mathcal{H}$ be a ...
3
votes
1answer
184 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
2
votes
0answers
119 views

A doubt about tensor product on Hilbert Spaces

An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$. Let ...
1
vote
0answers
40 views

Radial and angular part of the space of compactly supported smooth functions

On page 124 of Thaller's The Dirac equation the following space is mentioned : $$C^{\infty}_0(0, \infty)\otimes C^\infty(\mathbb{S}^2)\subset L^2(0, \infty)\otimes L^2(\mathbb{S}^2),$$ where the ...
1
vote
2answers
518 views

Inner product on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$, respectively. Then $H_1\otimes H_2$ is at least a pre-Hilbert space (we are ...
6
votes
2answers
225 views

Example of a non-algebraic $\ell^2$-function in two variables

Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
4
votes
1answer
290 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
12
votes
1answer
1k views

Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...