Linear dependence of two Hilbert space operators is expressed in terms of equality in modulus of certain sesquilinear and quadratic forms associated with the operators. The forms are based on generalized numerical ranges.

Linear dependence of two Hilbert space operators is expressed in terms of equality in modulus of certain sesquilinear and quadratic forms associated with the operators. The forms are based on generalized numerical ranges.

We prove that, for a Poisson vertex algebra $${\cal V}$$ , the canonical injective homomorphism of variational cohomology to its classical is an isomorphism, provided that viewed as differential algebra, polynomials in finitely many variables. This theorem one key ingredients computation cohomology. For proof, we introduce sesquilinear Hochschild and Harrison complexes vanishing symmetric