Matrix symbol of order two for some Banach algebras (Q5943609)

Two classes of Banach algebras which admit a matrix symbol of order 2 are described, and an explicit form of the symbol is obtained. Also we descibe all polynomials \(F(X,Y)\) and \(F(X,Y,Z)\) such that the norms of polynomials \(F(B,B^\ast)\) (when \(B^2=0\)) and \(F(S,W,W^\ast)\) (when \(S^2=I\), \(W^2=0\), \(S^\ast=S\)) depend only on the polynomial \(F\) and on the norms \(\|B\|\), \(\|W\|\). Applications to Banach algebras generated by singular integral operators and some their perturbations are considered.