3
votes
1answer
68 views

A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued ...
2
votes
1answer
83 views

Eigenvalues and Compact Resolvent

For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...
0
votes
0answers
83 views

Bounded operators with infinite matrix representations

I asked this question on StackExchange originally, but I'm giving it a go here as well. Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...
2
votes
1answer
152 views

Does a dense submodule of a free module always contain a basis?

Let $R$ be a completed normed ring, eg Banach algebra. Suppose that $F$ is a free $R$-module of infinite rank with a norm defined by the square root of sum of all norms of its components. If $F'$ is a ...
3
votes
1answer
757 views

Self-adjoint bounded operator, resolution of the identity, def. of the diagonal

Let $A$ be a self adjoint bounded linear operator with a continuous spectrum $\sigma(A)=[a,b]$ which acts on a separable Hilbert space. Let $E_\lambda$ be its resolution of the identity. For ...
9
votes
8answers
4k views

Can a self-adjoint operator have a continuous set of eigenvalues?

This should be a trivial question for mathematicians but not for typical physicists. I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...
0
votes
2answers
347 views

“Frobenius-finite” linear operators on a Hilbert Space

Let $H = L_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix ...
2
votes
3answers
701 views

Norm on quotient algebra of a tensor algebra

Suppose you have a finite dimensional real Hilbert space $V$ and you form the tensor algebra $$T(V) = \mathbb{R} \oplus V \oplus (V\otimes V) \oplus (V\otimes V \otimes V) \oplus \cdots$$ where the ...
8
votes
5answers
714 views

Abelianization of GL(H)

This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups. I am interested in a specific (infinite-dimensional) case of the above question. Let H be an ...