# zbMATH — the first resource for mathematics

Regular and singular J-inner matrix-functions and the corresponding interpolation problems. (English. Russian original) Zbl 0661.30029
Funct. Anal. Appl. 22, No. 1, 46-48 (1988); translation from Funkts. Anal. Prilozh. 22, No. 1, 57-59 (1988).
For J a given signature matrix $$(J=J^*=J^{-1})$$, a matrix function W meromorphic on E (when E is either the unit disk D or the upper half plane $$\pi)$$ is said to be J-inner if W has J-contractive values on E and J-unitary a.e. existing boundary values on the boundary of E. Such a function W is said to be a singular J-inner matrix function if both W and $$W^{-1}$$ can be expressed as the product of a bounded (and analytic on E) matrix function with the inverse of a bounded scalar outer function.
A J-inner function W is said to be regular if it has no nonconstant right singular divisor in the class of J-inner matrix functions. When $$J=I$$, all inner functions are regular, but for a general J, even rational nonconstant singular J-inner functions exist; such functions necessarily have all poles on the boundary of E and are called Brune sections in circuit theory.
J-inner functions arise naturally as the provider of a linear fractional parametrization of all solutions of a Nevanlinna-Pick-type interpolation problem; singular J-inner functions arise when the interpolation problem involves prescribed values for the function and its angular derivative on the boundary of E. The present paper obtains an explicit function theoretic characterization of singular J-inner functions, presents some elementary properties of the singular-regular factorization for J-inner functions, and solves the converse problem for finding the data for an interpolation problem in terms of a given J-inner function which parametrizes the solution set. All these results are for general (not necessarily rational) J-inner matrix functions.
Reviewer: J.A.Ball

##### MSC:
 30E05 Moment problems and interpolation problems in the complex plane 46E40 Spaces of vector- and operator-valued functions 47B50 Linear operators on spaces with an indefinite metric
Full Text:
##### References:
 [1] D. Z. Arov, Regular j-Inner Matrix-Functions and the Corresponding Continuation Problems [in Russian], Deposited at Ukrainskii NIINTI, No. 406-Uk87 (1987). [2] D. Z. Arov, ?-Generating Matrices, j-Inner Matrix-Functions and Related to Them Problems of Extrapolation of Matrix-Functions [in Russian], Deposited at Ukrainskii NIINTI, No. 726-Uk86 (1986). [3] D. Z. Arov, Lect. Notes Math.,1043, 164-168 (1984). [4] D. Z. Arov, Operator Theory: Advances and Applications, Birkh?user Verlag, Basel,24, 17-27 (1987). [5] I. V. Kovalishina and V. P. Potapov, Dokl. Akad. Nauk Arm. SSR, Ser. Mat.,9, No. 1, 3-9 (1974). [6] I. P. Fedchina, Dokl. Akad. Nauk Arm. SSR, Ser. Mat.,11, No. 4, 214-218 (1975). [7] D. Z. Arov and M. G. Krein, Funkts. Anal. Prilozhen.,15, No. 2, 61-64 (1981). · Zbl 0464.51002 · doi:10.1007/BF01082383 [8] D. Z. Arov and M. G. Krein, Acta Sci. Math.,45, 33-50 (1983). [9] J. A. Ball and J. W. Helton, J. Oper. Theory,9, 107-142 (1983). · Zbl 0505.47029 [10] J. W. Helton, Bull. Am. Math. Soc.,7, No. 1, 1-64 (1982). · Zbl 0493.46047 · doi:10.1090/S0273-0979-1982-15001-7 [11] A. A. Nudel’man, Dokl. Akad. Nauk SSSR,256, No. 4, 790-793 (1981). [12] D. Z. Arov, Sib. Mat. Zh.,20, No. 2, 211-228 (1979). [13] M. S. Livshits, Operators, Oscillations, Waves [in Russian], Nauka, Moscow (1966). [14] L. A. Simakova, Mat. Issled., Kishinev,10, No. 1, 287-292 (1975). [15] M. G. Krein and Sh. N. Saakyan, Funkts. Anal. Prilozhen.,4, No. 3, 103-104 (1970). [16] V. E. Katsnel’son, Methods of j-Theory in Continuous Interpolation Problems of Analysis [in Russian], Part I, Deposited at Vsesoyuznyi Institut Nauchnoi i Tekhnicheskoi Informatsii, No. 171-83 (1983). [”J-theory,” T. Anso, Hokkaido University, Sapporo (1985).] [17] L. A. Sakhnovich, Usp. Mat. Nauk,41, No. 1, 3-55 (1986). [18] M. G. Krein and H. Langer, J. Oper. Theory,13, 299-417 (1985). [19] V. P. Potapov, Tr. Mosk. Mat. Obshch.,4, 125-236 (1955). [20] D. Z. Arov and L. A. Simakova, Mat. Zametki,19, No. 4, 491-500 (1976).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.