## oa.operator algebras – Schaten p norm of block matrices

Let $$A=Doplus 0$$ be a diagonal Hermitian matrix and $$B$$ is an invertible Hermitian matrix with $$(1,1)$$ block being $$B_{11}$$ and $$B_{11}$$ and $$D$$ have the same dimensions. Then is it true that if $$(1+|t|^p)^{q/p}=|A+tB|_q^q$$ for all $$tinmathbb R,$$ then $$B_{12}=0$$ where $$B_{12}$$ is in the $$(1,2)$$ block of $$B$$?

$$|.|_q$$ denotes the Schatten-q norm and $$1

I somehow think that the answer is yes but cannot really prove it.

## oa.operator algebras – Lower bounds in the space of compact operators

Let $$H$$ be a separable Hilbert space, and $$K(H)$$ the corresponding space of compact operators. Consider the “unit sphere” $$S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$$. Is it true that, given any pair of operators $$T_1,T_2in S$$, there exists another operator $$Tin S$$ such that $$Tleq T_1,T_2$$?.

## Algebras oa.operator – Equivalence of the \$ sigma-weak topology with respect to another topology

Let $$mathcal H$$ to be a Hilbert space. Define a topology $$tau_1$$ sure $$B ( mathcal H)$$ by the seminorms family $$x mapsto | Tr (xa) |,$$ $$a in L ^ 1 (B ( mathcal H)).$$ Right here $$B ( mathcal H)$$ refers to the set of all linear maps delineated on $$mathcal H$$ and $$L ^ 1 (B ( mathcal H))$$ refers to the operators of the trace class.
Again define the $$sigma$$Topology -WOT $$tau_2$$ sure $$B ( mathcal H)$$ by removing the weak operator topology from $$B ( mathcal H otimes ell_2)$$ at $$B ( mathcal H)$$ via the map $$x mapsto x otimes 1.$$ How to show that $$tau_1 = tau_2$$? In many books and lecture notes in von Neumann's algebras, they have just mentioned that this is true. But I could not find any solid proof.