Representable Projections and Semi-Projections in a Hilbert Space

• Published in: Complex analysis and operator theory Vol. 15; no. 4
• Main Author: Labrousse, J. -Ph
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2021; Springer Nature B.V
• Subjects: Analysis; Hilbert space; Mathematics; Mathematics and Statistics; Operator Theory; Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo; 47A06; Projections; Semi-projections; Matrix representation
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-021-01092-9

Let H = H ⊕ 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 H = H 1 ⊕ H 2 and of K = K 1 ⊕ 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).