HOMOGENEOUS FUNCTIONS OF REGULAR LINEAR AND BILINEAR OPERATORS 1 To

This paper is a continuation of [5]. We apply the upper envelope representation method (or the quasilinearization method) in vector lattices developed in [4, 5] to the homogeneous functional calculus of linear and bilinear operators. Explicit formulae for computing φ̂(T1, . . . , TN ) for any finite sequence of regular linear or bilinear operators T1, . . . , TN are derived. For the theory of vector lattices and positive operators we refer to the books [1] and [3]. All vector lattices in this paper are real and Archimedean. Consider conic sets C and K with K ⊂ C and K closed. Let H (C;K) denotes the vector lattice of all positively homogeneous functions φ : C → R with continuous restriction to K. The expression φ̂(x1, . . . , xN ) can be correctly defined provided that the compatibility condition [x1, . . . , xN ] ⊂ K is hold, see [5]. Denote by H∨(R ,K) and H∧(R ,K) respectively the sets of all lower semicontinuous sublinear functions φ : RN → R ∪ {+∞} and upper semicontinuous superlinear functions ψ : RN → R∪{−∞} which are finite and continuous on a fixed cone K ⊂ RN . Put H∨(R ) := H∨(R N , {0}) and H∧(R ) := H∧(R , {0}). Denote by G∨(R ,K) and G∧(R ,K) respectively the sets of all lower semicontinuous gauges φ : RN → R+ ∪ {+∞} and upper semicontinuous co-gauges ψ : RN → R+ ∪ {−∞} which are finite and continuous on a fixed cone K ⊂ RN . Put G∨(R ) := G∨(R , {0}) and G∧(R N ) := G∧(R N , {0}). Observe that G∨(R ) ⊂H∨(R ) and G∧(R ) ⊂H∧(R ), see [4, 5]. Everywhere below E, F , and G denote vector lattices, while Lr(E,F ) and BLr(E,F ;G) stand for the spaces of regular linear operators from E to F and regular bilinear operator from E × F to G, respectively.