A Characterization of Generalized Zeros of Negative Type of Functions of the Class N_κ

Abstract

Recall ([1], [2], [3]) that N_κ denotes the set of all complex valued functions Q which are meromorphic in the open upper half plane C ⁺ and such that the kernel N_Q:

$${N_Q}\left( {z,\zeta } \right):\left( {Q\left( z \right) - \overline {Q\left( \zeta \right)} } \right)/\left( {z - \overline \zeta } \right)$$

(1.1)

for z,ζ ε D _Q has κ negative squares (here D _Q (⊂C ⁺) denotes the domain of holomorphy of Q). This means that for arbitrary n ε Z and z₁,z₂,...,z_n ε D _Q the matrix (N_Q(z_i,z_j)) ⁿ₁ has at most κ negative eigenvalues and for at least one choice of n, z₁,...,z_n it has exactly κ negative eigenvalues. The class N_o coincides with the Nevanlinna class of all functions which are holomorphic in C ⁺ and map C ⁺ into C ⁺ UR. The following two examples of functions of the class N₁ were considered in [2], [4], respectively:

$$w\left( z \right):\alpha - z + \int\limits_{ - \infty }^\infty {\left( {{{\left( {t - z} \right)}^{ - 1}} - t{{\left( {1 + {t^2}} \right)}^{ - 1}}} \right)} d{\sigma _O}\left( t \right),v\left( z \right): = \alpha + \left( {1/z} \right) + \int\limits_{ - 8}^\infty {\left( {{{\left( {t - z} \right)}^{ - 1}} - t{{\left( {1 + {t^2}} \right)}^{ - 1}}} \right)} d{\sigma _1}\left( t \right),$$

(1.2)

where α ε R and σ_o, σ_l are nondecreasing functions on R such that

$${\int\limits_{ - \infty }^\infty {\left( {1 + {t^2}} \right)} ^{ - 1}}d{\sigma _j}\left( t \right) < \infty ,j = 0,1\,{\sigma _1}\left( {0 + } \right) = {\sigma _1}\left( {0 - } \right).$$