Representable Projections and Semi-Projections in a Hilbert Space

Abstract

Let $H = {\mathcal H}\oplus {\mathcal K}$ be the direct sum of two Hilbert spaces. In this paper we characterise the semi-projections (defined in the paper) and projections with a given kernel and a given range that can be described by a two by two matrix or block of relations determined by the decompositions of ${\mathcal H}= {\mathcal H}_{1} \oplus {\mathcal H}_{2}$ and of ${\mathcal K}= {\mathcal K}_{1} \oplus {\mathcal K}_{2}$. This generalises the Stone - de Snoo (Oral communication to the author, 1992; J Indian Math Soc 15: 155–192, 1952) formula for the orthogonal projection on the graph of a closed linear relation, and extends the results of Mezroui (Trans AMS 352: 2789–2800, 1999) on the same subject. This requires some new results concerning blocks of linear relations as studied in (Adv Oper Theory 5: 1193–1228, 2020). Some applications are given on the product of two relations including one contained in (Complex Anal Oper Theory 6: 613–624, 2012).