On (m,p)(m,p)-expansive and (m,p)(m,p)-contractive operators on Hilbert and Banach spaces

For a bounded operator T on a Banach space X, we defineβ(m,p)(T,x):=∑k=0m(−1)m−k(mk)‖Tkx‖pfor all x∈X. We prove β(m,p)(T,x)≤0β(m,p)(T,x)≤0 for all x∈Xx∈X implies β(m−1,p)(T,x)≥0β(m−1,p)(T,x)≥0 for all x∈Xx∈X. This result allows us to extend several results for (m,p)(m,p)-isometries (β(m,p)(T,x)=0β(m,p)(T,x)=0) to operators only satisfying β(m,p)(T,x)≤0β(m,p)(T,x)≤0. A Berger–Shaw type result is also proved for a class of m  -expansive operators on a Hilbert space. Several classes of operators are introduced and studied by using β(m,p)(T,x)β(m,p)(T,x) in parallel with corresponding classes of operators on Hilbert space. In particular a class of p-subnormal operators on a Banach space is introduced and such operators are proved to be paranormal.