Noboru Okazawa

Logarithms and imaginary powers of closed linear operators

The imaginary powersAit of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC0-group {exp(itlogA);t ∈R} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) − log(1+A−1). LetA be a linearm-sectorial operator of typeS(tan ω), 0≤ω≤(π/2), in a Hilbert spaceX. That is, |Im(Au, u)| ≤ (tan ω)Re(Au, u) foru ∈D(A). Then ω±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC0-group {(1+A)it;t ∈R} of bounded imaginary powers, satisfying the estimate ‖(1+A)it‖ ≤ exp(ω|t|),t ∈R. In particular, ifA is invertible, then ω±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)−log(1+A−1), and {Ait;t ∈R} forms aC0-group onX, with the estimate ‖Ait‖ ≤ exp(ω|t|),t ∈R. This yields a slight improvement of the Heinz-Kato inequality.

Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials

The wellposedness of nonlinear Schrödinger equations (NLS) with inverse-square potentials is discussed in this article. The usual (NLS) is regarded as the potential-free case. The wellposedness of the usual (NLS) is well-known for a long time. In fact, several methods have been developed up to now. Among others, the Strichartz estimates seem to be essential in addition to the restriction on the nonlinear term caused by the Gagliardo–Nirengerg inequality. However, a parallel argument is not available when we apply such estimates to (NLS) with inverse-square potentials. Thus, we shall give only partial answer to the question in this article.

Logarithmic Characterization of Bounded Imaginary Powers

Let A be a closed linear operator with domain D (A) and range R (A) in a Banach space X. Our basic assumption consists of three conditions:

Semilinear Elliptic Problems Associated with the Complex Ginzburg-Landau Equation

\rho ( - A) \supset {{R}_{ + }} {\text{and}} \exists M \geqslant 1 {\text{such}} {\text{that}}\parallel \xi {{{\text{(}}A + \xi )}^{{ - 1}}}\parallel \leqslant M\forall \xi > 0.

An L p theory for Schrödinger operators with nonnegative potentials

(i)

On the perturbation of linear operators in Banach and Hilbert spaces

\overline {D(A)} = X.