Math Problems Vault

Problem 116

The restriction of the adjoint to the set of normal operators is strongly continuous.

?

Problem 117

If $\left\{ A_n\right\}$ and $\left\{ B_n\right\}$ are nets that converge strongly to $0$ , and if $\left\{ \left\| B_n\right\| \right\}$ is bounded, does it follow that $\left\{ A_nB_n\right\}$ converges strongly to $0$ ?

?

Problem 118

If $\left\{ f_n\right\}$ is a sequence of vectors and $\left\{ A_n\right\}$ is a sequence of operators such that $f_n \to f$ weakly and $A_n \to A$ strongly, does it follow that $A_nf_n \to Af$ weakly?

?

Problem 119

Is the spectrum strongly upper semicontinuous? What can be said about the spectral radius?

?

Remark.

?

The spectrum of an operator varies upper semicontinuously with respect to the norm topology (Problem 103): if an operator is replaced by one that is near it in the norm topology, then the spectrum can increase only a little. The question here is what happens to this semicontinuity when the strong topology is used in place of the norm topology -- that is, whether $A_n \to A$ (strongly) implies that $\operatorname {spec} A$ cannot be much larger, in a suitable limiting sense, than the spectra $\operatorname {spec} A_n$ .

Problem 120

Is a bounded increasing sequence of Hermitian operators necessarily strongly convergent? uniformly convergent?

?

Problem 121

If $A$ is an operator such that $0 \leq A \leq 1$ , and if a sequence $\left\{ B_n\right\}$ is defined recursively by the equations

B_0 = 0 \quad \text{and} \quad B_{n+1} = \tfrac {1}{2}\left((1-A) + B_n^{2}\right), \qquad n = 0, 1, 2, \ldots ,

then the sequence $\left\{ B_n\right\}$ is strongly convergent. If $\lim_n B_n = B$ , then $(1-B)^{2} = A$ .

?

Problem 122

If $E$ and $F$ are projections with ranges $\mathbf{M}$ and $\mathbf{N}$ , find the projection $E \wedge F$ with range $\mathbf{M} \cap \mathbf{N}$ .

?

Remark.

?

The problem is to find an "expression" for the projection described. Although most mathematicians would read the statement of such a problem with sympathetic understanding, it must be admitted that rigorously speaking it does not really mean anything. The most obvious way to make it precise is to describe certain classes of operators by the requirement that they be closed under some familiar algebraic and topological operations, and then try to prove that whenever $E$ and $F$ belong to such a class, then so does $E \wedge F$ . The most famous and useful classes pertinent here are the von Neumann algebras (called "rings of operators" by von Neumann). A von Neumann algebra is an algebra of operators (i.e., a collection closed under addition and multiplication, and closed under multiplication by arbitrary scalars), self-adjoint (i.e., closed under adjunction), containing $1$ , and strongly closed (i.e., closed with respect to the strong operator topology). For von Neumann algebras, then, the problem is this: prove that if a von Neumann algebra contains two projections $E$ and $F$ , then it contains $E \wedge F$ .

Reference: [150, vol.2, p.55].