Factorizations, invariant subspaces and multivalency H.L. (Rudi) Wietsma Department of Mathematics and Statistics, University of Vaasa December 19, 2019 Partly joint work with Seppo Hassi H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 1 / 19
Motivation Motivation (1) The difference between the factorization of operator-valued and scalar-valued generalization Nevanlinna functions. In the scalar case the factorization reflects precisely the spectral properties of the underlying minimal realizing self-adjoint operator. If the generalized Nevanlinna function f has the factorization � κ i =1 ( z − α i )( z − α i ) f ( z ) = f 0 ( z ) , � κ i =1 ( z − β i )( z − β i ) where f 0 is an ordinary Nevanlinna function, then the eigenvectors of nonpositive type of any selfadjoint relation A in any minimal operator realization for f , I + ( z − z 0 )( A − z ) − 1 � �� � f ( z ) = c + iz 0 [ ω, ω ] + ( z − z 0 ) ω, ω , reflect this factorization: Abnormal poles of f (the β i ) are eigenvalues of A whose eigenvectors are nonpositive; In the operator case this is no longer true for known factorizations, see Luger’s A factorization of regular generalized Nevanlinna functions . (2) The difference between factorizing generalized Nevanlinna functions and generalized Schur functions. Why was the factorization of generalized Schur functions easily established, while the factorization of generalized Nevanlinna function was much harder to establish? H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 2 / 19
An operator-theoretical approach An operator-theoretical approach The difficulty of establishing the factorization of generalized Nevanlinna functions is reflected by their definition: Definition f is a generalized Nevanlinna function of index κ ∈ N , or f ∈ N κ for short, when f is symmetric, meromorphic on C \ R and its Nevanlinna kernel : N f ( z , w ) = f ( z ) − f ( w ) ∗ , z , w ∈ D ( f ) , z − w has κ negative squares. This means that for arbitrary n ∈ N , z 1 , . . . , z n ∈ D ( f ) ∩ C + and � � ξ 1 , . . . , ξ n ∈ H , the Hermitian matrix N f ( z i , z j ) i , j =1 ,... n has at most κ negative eigenvalues, and κ is minimal with this property. This definition is operator-theoretical. Practically one starts from the following characterization: Theorem (Realization property) If A is a selfadjoint relation with non-empty resolvent set in a Pontryagin space { Π , [ · , · ] } , then f ( z ) defined for z ∈ ρ ( A ) and arbitrary, but fixed, z 0 ∈ ρ ( A ) by I + ( z − z 0 )( A − z ) − 1 � f ( z ) := c + iz 0 [ ω, ω ] + ( z − z 0 ) �� ω, ω � , where c ∈ R and ω ∈ Π , is a generalized Nevanlinna function whose index is at most the negative index of { Π , [ · , · ] } . Conversely, if f ∈ N κ , then there exists a selfadjoint relation A with non-empty resolvent set in a Pontryagin space { Π , [ · , · ] } , whose negative index is (precisely/at least) κ , such that the preceding identity holds for some ω ∈ Π and c ∈ R . H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 3 / 19
An operator-theoretical approach Therefore, to prove the factorization property of generalized Nevanlinna functions, Theorem (Factorization property) A complex function f is a generalized Nevanlinna function of index κ if and only if there exists an ordinary Nevanlinna function f 0 such that � κ i =1 ( z − α i )( z − α i ) f ( z ) = f 0 ( z ) . � κ i =1 ( z − β i )( z − β i ) it would be expected that one has to use properties of self-adjoint relations in Pontryagin spaces. It turns out that the following property is needed. Theorem (Weak invariant subspace property) Let A be a selfadjoint relation (multi-valued operator) with non-empty resolvent set in a Pontryagin space { Π , [ · , · ] } with nonzero negative index κ . Then there exists a λ ∈ C ∪ {∞} for which ker ( A − λ ) contains a non-trivial nonpositive vector. This property can be easily seen to be equivalent to the normal (strong) invariant subspace property. H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 4 / 19
An operator-theoretical approach Leaving aside the special case of Delsarte, Genin and Kamp (1986) (closely connected to the Inner-Outer factorization), the first proofs of the factorization property of generalized Nevanlinna functions appeared in: (1) Derkach, Hassi & Snoo (1999); (2) Dijksma, Langer & Luger (2000). Both proofs made use of the characterization of the index of generalized Nevanlinna functions in terms of the growth of the function. More precisely, via the concept of the generalized pole of non-positive type (GPNT). However, that characterization had been obtained in Krein & Langer 1981 from the invariant subspace property of contractive operators in Pontryagin spaces. The preceding discussion strongly hints that this intermediate step is unnecessary; i.e., that it is possible to establish the factorization directly from the invariant subspace property of selfadjoint relations in Pontryagin spaces. H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 5 / 19
An operator-theoretical approach Theorem (A realization model) Let f ∈ N κ not be identically equal to zero and let { A , ω } realize f minimally (in particular, let the realizing space { Π , [ · , · ] } have negative index κ ). Then there exists e ∈ R such that f r ( z ) := ( z − α )( z − α ) f ( z ) , α, β ∈ C , α � = β and α � = β ( z − β )( z − β ) is realized by { A r , ω r } , where [ · , ω ] − e β . 0 A A r = ω 0 0 β Moreover, the realizing space { Π r , [ · , · ] r } corresponding to { A r , ω r } is the Pontryagin space { Π r , [ · , · ] r } with negative index κ + 1 defined via [ g , h ] r := [ g c , h c ] + g r h l + g l h r , g = { g l , g c , g r } , h = { h l , h c , h r } ∈ Π r := C × Π × C . Additionally, if there exist x , � x ∈ Π \ { 0 } such that [ x , x ] ≤ 0 , [ � x , � x ] ≤ 0 , { x , α x } ∈ A and x } ∈ � { � x , β � A, then f r ∈ N κ − 1 . H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 6 / 19
An operator-theoretical approach Proving the invariance property when assuming the factorization Let A be a self-adjoint relation in some Pontryagin space Π := { [ · , · ] } with negative index κ and let ω ∈ Π be such that [ ω, ω ] ≤ 0. Then by the realization property the pair { A , ω } realizes a generalized Nevanlinna function f ( z ). If this realization proves to be a non-minimal realization, then it is easily established that A possesses the (weak) invariance property. On the other hand, if it is a minimal realization, then f ∈ N κ and, hence, by the factorization property f ( z ) = ( z − α )( z − α ) f κ − 1 ( z ) , where f κ − 1 ∈ N κ − 1 . ( z − β )( z − β ) Making use of the preceding model another (possibly) non-minimal realization { A f , ω f } for f is obtained in which β ∗ ∗ ; A f = 0 A k − 1 ∗ 0 0 ∗ here A k − 1 is connected with any minimal realization for f k − 1 . Hence, { 1 } × { 0 } × { 0 } is a neutral A f invariant vector; i.e., it has the desired properties. (1) If this constructed realization is minimal, then the original realization (containing the arbitrary self-adjoint relation A ) is unitary equivalent to this one and, hence, A also has the weak invariant subspace property; (2) If this constructed realization is not minimal, then one has to be a bit of work to obtain the same result. H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 7 / 19
An operator-theoretical approach Proving the factorization when assuming the weak invariance property f := − f − 1 ∈ N κ . By the realization property Assume that f ∈ N κ , κ � = 0. Then also � there exists c ∈ C , a Pontryagin space { Π , [ · , · ] } with negative index κ , a selfadjoint relation A in Π and an element ω ∈ Π which together realize f ; c ∈ C , a selfadjoint relation � similarly, there exists � A in (the same Pontryagin space) ω ∈ Π which together realize � Π and an element � f . By the (weak) invariant subspace property there exist α, β ∈ R (for simplicity’s sake assume that α � = ∞ and β � = ∞ ) and x α , � x β ∈ Π such that x β } ∈ gr ( � { x α , α x α } ∈ gr ( A ) , [ x α , x α ] ≤ 0 , { � x β , β � A ) [ � x β , � x β ] ≤ 0 . and But then by the preceding theorem/model: z − α z − β f ( z ) z − α ∈ N κ − 1 . z − β Repeating the procedure establishes the factorization. H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 8 / 19
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