The notion of quadratic numerical range was introduced by H. Langer
and C. Tretter
[J. Oper. Theory 39, No. 2, 339–359 (1998; Zbl 0996.47006
)] and the concept of block numerical range was considered several years later by C. Tretter
and M. Wagenhofer
[SIAM J. Matrix Anal. Appl. 24, No. 4, 1003–1017 (2003; Zbl 1051.47003
)]. The aim of the present paper is to introduce the notion of total decomposition and to study its influence on the block numerical range.