Tagged Questions
4
votes
1answer
62 views
Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?
Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
2
votes
0answers
113 views
Must-read papers in Operator Theory
I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
1
vote
1answer
47 views
An explicit example of an invariant halfspace of the unilateral shift?
In a recent talk, A. Popov stated the following fact
The unilateral shift on $\ell^2$ has invariant halfspaces.
Halfspaces are closed subspaces whose dimension and codimension are both infinite.
...
1
vote
1answer
130 views
An upper bound for $\|(\lambda-A)^{-1}\|$?
Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$.
If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...
13
votes
1answer
277 views
How does $\sigma(T)$ change with respect to $T$?
Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane.
I wonder whether there is some result concerning how ...
2
votes
1answer
119 views
Subadditive, submultiplicative and scalar-multiplication-invariant functions
Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...
6
votes
1answer
162 views
Weak-* continuity of the adjoint map on a $W^*$-algebra
Let $\mathcal{M}$ be a $W^*$-algebra, i.e. a $C^*$-algebra with a Banach space predual $\mathcal{M}_*$. I'm trying to show that the adjoint map $x \mapsto x^*$ on $\mathcal{M}$ is weak-* (aka ...
7
votes
0answers
227 views
Motivation for abstract harmonic analysis
I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting.
However it seems Folland does not give many examples to illustrate the motivation behind much of ...
11
votes
1answer
322 views
Renorming $\mathcal{B}(\mathcal{H})$?
Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
