Spectral bounds derived from quadratic forms on decomposable tensors

AbstractThe problem we study concerns mn-by-mn real symmetric matrices A = [Aij]. The objective is to obtain best possible bounds on the spectrum of A given that the quadratic form on decomposable unit vectors in R has values restricted to the unit interval [0, 1]. We also discuss the relationships between our work and the theory of elliptic partial differential equations, especially to the Legendre-Hadamard condition. In particular, the examples we give to show that our bounds are best possible are related to a classical example in partial differential equations given by De Giorgi in 1968