This paper is the companion article to [Ann. Probab. 39 (2011) 779–856]. We consider a discrete elliptic equation on the d-dimensional lattice Zd with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (co...

We consider the Schrödinger map initial-value problem { ∂tφ = φ×∆φ on R × R; φ(0) = φ0, where φ : R × R → S →֒ R is a smooth function. In all dimensions d ≥ 2, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ∈ C(R : H∞ Q ), Q ∈ S, provided that the data φ0 ∈ H∞ Q is smooth and satisfies the smallness condition ‖φ0−Q‖Ḣd/2 ≪ 1. We prove also that th...

Let be a C∗-algebra with identity 1, and let s( ) denote the set of all states on . For p,q,r ∈ [1,∞), denote by r( ) the set of all infinite matrices A= [ajk]j,k=1 over such that the matrix (φ[A[2]]) [r] := [(φ(ajkajk))]j,k=1 defines a bounded linear operator from p to q for all φ∈ s( ). Then r( ) is a Banach algebra with the Schur product operation and norm ‖A‖ = sup{‖(φ[A[2]])‖ : φ∈ s( )}. A...

The present communication suggests deductive systems for the operator a of unprovability in some particular propositional normal modal logics. We give thus complete syntactic characterization of these logics in the sense of Lukasiewicz: for every formula φ either ` φ or a φ (but not both) is derivable. In particular, purely syntactic decision procedure is provided for the logics under considera...

Let BN be the unit ball in the N-dimensional complex space, for ψ, a holomorphic function in BN , and φ, a holomorphic map from BN into itself, the weighted composition operator on the weighted Hardy space H(β,BN ) is given by (Cψ,φ) f = ψ(z) f (φ(z)), where f ∈H2(β,BN ). This paper discusses the spectrum of Cψ,φ when it is compact on a certain class of weighted Hardy spaces and when the compos...

If φ is an analytic map of the unit diskD into itself, the composition operator Cφ on the Hardy space H (D) is defined by Cφ(f) = f ◦ φ. For a certain class of composition operators with multivalent symbol φ, we identify a subspace of H(D) on which C∗ φ behaves like a weighted shift. We reproduce the description of the spectrum found in [5] and show for this class of composition operators that ...

In 1884, G. Koenigs solved Schroeder’s functional equationf ◦ φ = λf in the following context: φ is a given holomorphic function mapping the openunit disk U into itself and fixing a point a ∈ U , f is holomorphic on U , and λis a complex scalar. Koenigs showed that if 0 <|φ′(a)| < 1, then Schroeder’sequation for φ has a unique holomorphic solution σ satisfyingσ ◦ φ =...

Some asymptotic properties of principal solutions of the halflinear differential equation (*) (a(t)Φ(x′))′ + b(t)Φ(x) = 0 , Φ(u) = |u|u, p > 1, is the p-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (1),...