In this paper we consider various aspects of generalized invertibility of the operator matrixacting on a Banach space$X$⊕$Y$.

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.

In this article, we classify all standard invariants that can arise from a composed inclusion of an A3with an A4subfactor. More precisely, if N⊂P is an A3subfactor and P⊂M is an A4subfactor, then only four standard invariants can arise from the composed inclusion N⊂M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A4subfactors.

We discuss the boundary behavior of functions in star-invariant subspaces ($BH$^{2})^{⊥}, where$B$is a Blaschke product. Extending some results of Ahern and Clark, we are particularly interested in the growth rates of functions at points of the spectrum of$B$where$B$does not admit a derivative in the sense of Carathéodory.

It is proved that the natural embedding of a separable Banach space$X$into the corresponding Bourgain–Pisier space extends $\mathcal{L}_{\infty}$ -valued operators.