# Analysis, Geometry, and Dynamical Systems Seminar

## Next session

### Reflexivity of bilattices

My talk will be devoted to reflexivity and hyperreflexivity of bilatties.  Bilattices were defined by Shulman in [3]. These structures were studied later in [4] in connection with operator synthesis, in [3] in the context of reflexivity and in [2] in the context of hyperreflexivity. A subspace analogue for a lattice is called a bilattice [4]. Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces. Denote by $\mathcal{P}(\mathcal{H})$ the lattice of all orthogonal projections on $\mathcal{H}$.  A bilattice is a set $\Sigma\subseteq\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$ containing pairs $(0,I), (I,0), (0,0)$ and $(P_1\wedge P_2, Q_1\vee Q_2), (P_1\vee P_2, Q_1\wedge Q_2)\in\Sigma$ whenever $(P_1,Q_1), (P_2,Q_2)\in\Sigma$.

Define after [5] $\operatorname{op}\Sigma = \{T\in\mathcal{B}(\mathcal{H},\mathcal{K}) : QTP = 0, \ \forall (P,Q)\in\Sigma\}.$ Then $\operatorname{op}\Sigma$ is always a reflexive subspace and all reflexive subspaces are of this form. The bilattice $\operatorname{bil}\mathcal{S}$ of a subspace $\mathcal{S}\subseteq\mathcal{B}(\mathcal{H},\mathcal{K})$ is defined to be the set $\operatorname{bil}\mathcal{S} = \{(P,Q) : Q\mathcal{S} P = \{0\}\}.$ A bilattice  $\Sigma$ is called reflexive if $\operatorname{bil}\operatorname{op}\Sigma = \Sigma$.

Given a bilattice $\Sigma\subseteq \mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$ and a pair of projections $(P,Q)\in\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$, let $\alpha((P,Q),\Sigma) = \sup\{\|QTP\| : \|T\|\leq 1, T\in\operatorname{op}\Sigma\}$ and $d((P,Q),\Sigma)=\inf\{\|P-L_1\|+\|Q-L_2\|: (L_1,L_2)\in\Sigma\}.$ A bilattice $\Sigma\subseteq\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$ is called hyperreflexive if there exists a constant $\kappa > 0$ such that $d((P,Q), \Sigma)\leq \kappa \alpha((P,Q),\Sigma)$, for each pair $(P,Q)\in\mathcal{P}(\mathcal{H})\times\mathcal{P} (\mathcal{K})$.

### References

1. K. R. Davidson and K. J. Harrison, Distance formulae for subspace lattices, J. London Math. Soc. (2) 39 (1989), 309-323.
2. K. Klis-Garlicka, Hyperreflexivity of bilattices, Czech. Math. J. 66(141) (2016), 119-125.
3. K. Klis-Garlicka, Reflexivity of bilattices, Czech. Math. J. 63(138) (2013), 995-1000.
4. V. S. Shulman, in "Nest Algebras" by  K. R. Davidson: a review, Algebra and Analiz, vol.2, no. 3 (1990), 236-255.
5. V. S. Shulman and L. Turowska, Operator synthesis. I. Synthetic sets, bilattices and tensor algebras, J. Functional Analysis, 209 (2004), no. 2, 293-331.