Difference between revisions of "BeurlingLax theorem"
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A theorem involved with the characterization of shiftinvariant subspaces of the Hardy space of analytic functions on the unit disc in terms of inner functions (cf. also Hardy spaces). Specifically, the space can be characterized as the space of analytic functions on the unit disc having Taylor series representation
with squaresummable coefficients (i.e., ). The shift operator on is defined to be the operator of multiplication by the coordinate function : . A closed linear subspace of is said to be invariant for if whenever . In 1949 (see [a6]), A. Beurling proved that any such shiftinvariant subspace has the form
where is an inner function. Here, an inner function is an analytic function on the unit disc with contractive values such that its boundary values (which exist for almost every point with respect to Lebesgue measure on the unit circle) have modulus (i.e. ) for almost all . The usefulness of the result is enhanced by the fact that such inner functions can be factored , where is a Blaschke product which collects all the zeros of in the unit disc, and a singular inner function , given by an explicit integral formula in terms of a singular measure on the unit circle. When one applies the result to the cyclic invariant subspace generated by a given function in , one arrives at the innerouter factorization of . Here, the right factor is an outer function; such functions are characterized operatortheoretically as the cyclic vectors for the operator , or functiontheoretically by an integral representation formula arising from an absolutely continuous measure on the unit circle. P.D. Lax [a11] extended the result to finitedimensional vectorvalued (in the alternative setting where of the unit disc is replaced by of the right halfplane). Later, P.R. Halmos [a12] gave an elegant proof for the infinitedimensional case. V.P. Potapov [a15] worked out an analogue of the parametrization (in terms of zeros and a singular measure on the circle) of an inner function for the matrixvalued case. It turns out that the parametrization of matrix inner functions (and, more generally, of contractive analytic matrixfunctions on the unit disc) in terms of Schur parameters (see [a9] and [a16]) has proved more useful for engineering applications.
Generalizations and applications of this result have been going strong (with some interruptions) to the present day (1998). Beurling already observed that the detailed parametrization of inner functions (in terms of zero locations and singular measures on the unit circle as sketched above) leads to a complete characterization of the lattice structure of the lattice of invariant subspaces for the shift operator (a strikingly different structure from the previously worked out case of selfadjoint or unitary operators). In the 1960s, operator theorists realized that the compression of the shift operator to the orthogonal complement of a shiftinvariant subspace serves as a model for a rich class of Hilbertspace operators. The model theories of L. de Branges and J. Rovnyak [a7] and that of B. Sz.Nagy and C. Foiaş [a18] give two roughly equivalent generalizations of , each of which leads to a model for an arbitrary, completely nonunitary contraction operator on a Hilbert space. In both of these theories, the function (a contractive, operatorvalued, analytic function on the unit disc) is called the characteristic function of the associated operator . In the case where is inner (i.e., the boundary values on the unit circle are isometries almost everywhere), the model space reduces to the Beurling–Lax form . There also has evolved a theory of socalled operators, for which the operator (with a scalar inner function) are the building blocks in an analogue of a canonical Jordan form which classifies operators up to quasisimilarity (see [a18] and [a5]).
The model theoretical approach of M.S. Livsic and coworkers in the former Soviet Union (see [a8]) gives an alternative formulation with emphasis on the systemtheoretic aspects (where the characteristic function appears as the transfer function of a discretetime, linear, energyconserving system) rather than on connections with Beurling–Lax representations for shiftinvariant subspaces. In 1983, J.A. Ball and J.W. Helton [a3] showed how a new type of Beurling–Lax representation for the vectorvalued case (where one demands that and preserve an indefinite rather than Hilbertspace inner product) leads directly to the linearfractional parametrization for the solution set of a Nevanlinna–Pick interpolation problem. Connections with engineering applications, such as signal processing, robust control and system identification, have led to new questions, different points of view, and an emphasis on robust computational procedures; in particular, the model space can be interpreted as the range of a controllability operator, and there are explicit procedures for realizing as the transfer function of a unitary system
from the common zeros of the invariant subspace (see, e.g., [a2] and [a16]). There have been recent extensions of a number of these ideas to timevarying systems (see, e.g., [a10]) and even to nonlinear systems (see [a4]).
There have been other extensions from the point of view of functiontheoretic operator theory. Already in the 1960s, other workers (such as D. Lowdenslager, H. Helson and J. Wermer) obtained Beurlingtype representations for shiftinvariant subspaces in or , as well as extensions to more abstract harmonic analysis and functionalgebra settings. A more delicate Beurling–Lax representation theorem has been shown to hold for Hardy spaces on finitelyconnected planar domains, where unitary representations of the fundamental group of the underlying domain play a fundamental role in the description (see, e.g., [a1]). Recently (1990s) there have been attempts to generalize Halmos' wandering subspace construction in still other directions. For example, there is a Beurlinglike representation theorem for invariant subspaces of the Dirichlet shift (see [a17]) and a notion of inner divisor for the Bergman space [a14] (cf. also Bergman spaces).
References
[a1]  M.B. Abrahamse, R.G. Douglas, "A class of subnormal operators related to multiply connected domains" Adv. Math. , 19 (1976) pp. 1–43 
[a2]  J.A. Ball, I. Gohberg, L. Rodman, "Interpolation of rational matrix functions" , Oper. Th. Adv. Appl. , 45 , Birkhäuser (1990) 
[a3]  J.A. Ball, J.W. Helton, "A Beurling–Lax theorem for the Lie group which contains most classical interpolation" J. Operator Th. , 9 (1983) pp. 632–658 
[a4]  J.A. Ball, J.W. Helton, "Shift invariant manifolds and nonlinear analytic function theory" Integral Eq. Operator Th. , 11 (1988) pp. 615–725 
[a5]  H. Bercovici, "Operator theory and arithmetic in " , Math. Surveys Monogr. , 26 , Amer. Math. Soc. (1988) 
[a6]  A. Beurling, "On two problems concerning linear transformations in Hilbert space" Acta Math. , 81 (1949) pp. 239–255 
[a7]  L. de Branges, J. Rovnyak, "Square summable power series" , Holt, Rinehart&Winston (1966) 
[a8]  M.S. Brodskii, "Triangular and Jordan representations of linear operators" , Transl. Math. Monogr. , 32 , Amer. Math. Soc. (1971) 
[a9]  "I. Schur Methods in Operator Theory and Signal Processing" I. Gohberg (ed.) , Oper. Th. Adv. Appl. , 18 , Birkhäuser (1986) 
[a10]  "Time–Variant Systems and Interpolation" I. Gohberg (ed.) , Oper. Th. Adv. Appl. , 56 , Birkhäuser (1992) 
[a11]  P.D. Lax, "Translation invariant subspaces" Acta Math. , 101 (1959) pp. 163–178 
[a12]  P.R. Halmos, "Shifts on Hilbert spaces" J. Reine Angew. Math. , 208 (1961) pp. 102–112 
[a13]  J.W. Helton, "Operator theory, analytic functions, matrices, and electrical engineering" , Conf. Board Math. Sci. , 68 , Amer. Math. Soc. (1987) 
[a14]  H. Hedenmalm, "A factorization theorem for square areaintegrable analytic functions" J. Reine Angew. Math. , 422 (1991) pp. 45–68 
[a15]  V.P. Potapov, "The multiplicative structure of contractive matrix functions" Amer. Math. Soc. Transl. , 15 : 2 (1960) pp. 131–243 
[a16]  P.A. Regalia, "Adaptive IIR filtering and signal processing and control" , M. Dekker (1995) 
[a17]  S. Richter, "A representation theorem for cyclic analytic twoisometries" Trans. Amer. Math. Soc. , 328 (1991) pp. 325–349 
[a18]  B. Sz.Nagy, C. Foias, "Harmonic analysis of operators on Hilbert space" , NorthHolland (1970) 
BeurlingLax theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BeurlingLax_theorem&oldid=17989