Strict isometric and strict symmetric commuting d-tuples of Banach space operators

Abstract

Given commuting d-tuples Si and Ti, 1 i 2, of Banach space operators
such that the tensor products pair (S1
S2;T1
T2) is strict m-isometric (resp.,
S1, S2 are invertible and (S1
S2;T1
T2) is strict m-symmetric), there exist
integers mi > 0, and a non-zero scalar c, such that m = m1 +m2 ????1, (S1; 1
cT1)
is strict m1-isometric and (S2; cT2) is strict m2-isometric (resp., there exist
integers mi > 0, and a non-zero scalar c, such that m = m1 +m2 ????1, (S1; 1
cT1)
is strict m1-symmetric and (S2; cT2) is strict m2-symmetric). However, (Si;Ti)
is strict mi-isometric (resp., strict mi-symmetric) for 1 i 2 implies only
that (S1
S2;T1
T2) is m-isometric (resp., (S1
S2;T1
T2) is m-symmetric).