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).

令\（H = {\ mathcal H} \ oplus {\ mathcal K} \）为两个希尔伯特空间的直接和。在本文中，我们用给定的核和给定的范围来刻画半投影（在本文中定义）和投影，这些投影可以用由2（2）分解或由\（{\ mathcal H } = {\ mathcal H} _ {1} \ oplus {\ mathcal H} _ {2} \）和\（{\ mathcal K} = {\ mathcal K} _ {1} \ oplus {\ mathcal K} _ {2} \）。这概括了闭合线性关系图上的正交投影的Stone-de Snoo（与作者进行口头交流，1992年； J Indian Math Soc 15：155–192，1952年）公式，并扩展了Mezroui（Trans AMS 352：2789-2800，1999）。这需要一些有关线性关系块的新结果，如Adv Oper Theory 5：1193–1228，2020中所述。在两种关系的乘积上给出了一些应用，其中一种包含在（复杂肛门操作理论6：613–624，2012）中。