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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 f (z) = κ

i=1(z − αi)(z − αi)

κ

i=1(z − βi)(z − βi)

f0(z), where f0 is an ordinary Nevanlinna function, then the eigenvectors of nonpositive type

• f any selfadjoint relation A in any minimal operator realization for f ,

f (z) = c + iz0[ω, ω] + (z − z0)

• I + (z − z0)(A − z)−1

ω, ω

• ,

reflect this factorization: Abnormal poles of f (the βi) 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 ∈ Nκ for short, when f is symmetric, meromorphic on C \ R and its Nevanlinna kernel: Nf (z, w) = f (z) − f (w)∗ z − w , z, w ∈ D(f ), has κ negative squares. This means that for arbitrary n ∈ N, z1, . . . , zn ∈ D(f ) ∩ C+ and ξ1, . . . , ξn ∈ H, the Hermitian matrix

• Nf (zi, zj)
• 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, z0 ∈ ρ(A) by f (z) := c + iz0[ω, ω] + (z − z0)

• I + (z − z0)(A − z)−1

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 ∈ Nκ, 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 f0 such that f (z) = κ

i=1(z − αi)(z − αi)

κ

i=1(z − βi)(z − βi)

f0(z). 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

• btained in Krein & Langer 1981 from the invariant subspace property of contractive
• perators 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 ∈ Nκ 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 fr(z) := (z − α)(z − α) (z − β)(z − β) f (z), α, β ∈ C, α = β and α = β is realized by {Ar, ωr}, where Ar =   β [·, ω] −e A ω β   . Moreover, the realizing space {Πr, [·, ·]r} corresponding to {Ar, ωr} is the Pontryagin space {Πr, [·, ·]r} with negative index κ + 1 defined via [g, h]r := [gc, hc] + grhl + glhr, g = {gl, gc, gr}, h = {hl, hc, hr} ∈ Πr := C × Π × C. Additionally, if there exist x, x ∈ Π \ {0} such that [x, x] ≤ 0, [ x, x] ≤ 0, {x, αx} ∈ A and { x, β x} ∈ A, then fr ∈ Nκ−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 ∈ Nκ and, hence, by the factorization property f (z) = (z − α)(z − α) (z − β)(z − β) fκ−1(z), where fκ−1 ∈ Nκ−1. Making use of the preceding model another (possibly) non-minimal realization {Af , ωf } for f is obtained in which Af =   β ∗ ∗ Ak−1 ∗ ∗   ; here Ak−1 is connected with any minimal realization for fk−1. Hence, {1} × {0} × {0} is a neutral Af 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

• btain 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

Assume that f ∈ Nκ, κ = 0. Then also f := −f −1 ∈ Nκ. By the realization property 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 ; similarly, there exists c ∈ C, a selfadjoint relation A in (the same Pontryagin space) Π and an element ω ∈ Π which together realize f . By the (weak) invariant subspace property there exist α, β ∈ R (for simplicity’s sake assume that α = ∞ and β = ∞) and xα, xβ ∈ Π such that {xα, αxα} ∈ gr(A), [xα, xα] ≤ 0, { xβ, β xβ} ∈ gr( A) and [ xβ, xβ] ≤ 0. But then by the preceding theorem/model: z − α z − β f (z)z − α z − β ∈ Nκ−1. Repeating the procedure establishes the factorization.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 8 / 19

An operator-theoretical approach

Intermediate conclusions

(1) The kernel characterization (definition) of generalized Nevanlinna functions was not used explicitly at all. (2) It has been established that the factorization and invariance property are equivalent in the sense that they imply each other (via the presented model). (3) This is a circular argument, but it implies that if a simple argument for the factorization is found, one obtains at the same time a simple argument for the invariant subspace property. (4) A relatively simple, and completely function-theoretical, argument for the factorization of generalized Nevanlinna is obtained (starting from the kernel definition!) via the concept of multivalency. This concept provides us with a truly function-theoretic definition of generalized Nevanlinna functions. (5) An even simpler proof is obtained when starting from the similar problem of factorizing generalized Schur functions.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 9 / 19

An operator-theoretical approach

Generalized Schur functions

Definition

A H-valued function S function meromorphic on D is an (operator valued) generalized Schur function with index κ ∈ N0, S ∈ Sκ(H), if its Schur kernel KS, defined as KS(z, w) = IH − S(z)S(w)∗ 1 − zw , has κ negative squares on D ∩ ρ(S), where ρ(S) is the domain of holomorphy of S.

Theorem (Kre˘ ın-Langer factorization, 1970)

S ∈ Sκ(H) if and only if there exists S0 ∈ S0(H) such that S(z) = κ

• i=1
• I − Pi + 1 − γiz

z − γi Pi

• S0(z)
• s(z) =

κ

• i=1

1 − γiz z − γi s0(z)

• γi ∈ D,

where the Pi are one-dimensional projections having certain properties.

Theorem (Invariant subspace property)

Let T be a contractive operator in a Pontryagin space {Π, [·, ·]} with negative index κ. Then there exists a κ-dimensional T-invariant subspace M such that {M, −[·, ·]} is a Hilbert space.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 10 / 19

An operator-theoretical approach

Remarks

(1) The Kre˘ ın-Langer factorization was established by combining an operator realization

• f (operator-valued) generalized Schur functions with the invariant subspace

property of contractions in Pontryagin spaces. (2) Similarly as in the Nevanlinna case, the Kre˘ ın-Langer factorization also, conversely, implies the invariant subspace property of contractions in Pontryagin spaces. (3) Different from the Nevanlinna-setting, the Kre˘ ın-Langer factorization for

• perator-valued generalized Schur functions can be straight-forwardly deduced from

its scalar-valued case. Because of the intimate connection between between generalized Schur and generalized Nevanlinna functions, a simple proof for all statements is obtained when the factorization

• f scalar-valued generalized Schur functions is established in a simple manner. This will

now be done by means of the concept of multivalency.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 11 / 19

A function-theoretical approach

Multivalency

A complex function f is at most, exactly or at least κ-valent, κ ∈ N0, with respect to (the set) A ⊆ C ∪ {∞} on (the set) ∆ ⊆ C if the multiplicities of the zeros of f (z) − w

• n ∆ ∩ D(f ) add up to at most, exactly, or at least κ for every fixed w ∈ A: this will be

written as V (f , ∆, A) ≤ κ, V (f , ∆, A) = κ or V (f , ∆, A) ≥ κ, respectively. Moreover, we use the notation Dc to denote the complement of clos(D) in C ∪ {∞}: Dc = {z ∈ C ∪ {∞} : |z| > 1}. Similarly, the notation C+ and C− is used to denote the upper and lower halfplanes: C+ = {z ∈ C : Im z > 0}; C− = {z ∈ C : Im z < 0}. Notice that the classes of ordinary Nevanlinna and ordinary Schur functions can be characterized by the concept of multivalency: f ∈ N0 ⇐ ⇒ V (f , C+, C−) = 0; s ∈ S0 ⇐ ⇒ V (s, D, Dc) = 0. We will show that the preceding can be generalized to: f ∈ Nκ ⇐ ⇒ V (f , C+, C−) = κ; s ∈ Sκ ⇐ ⇒ V (s, D, Dc) = κ.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 12 / 19

A function-theoretical approach

Proposition

Let s be meromorphic on D. Then equivalent are: (i) V (s, D, Dc) = κ; (ii) (a) there exists w ∈ Dc such that V (s, D, w) = κ; (b) lim supz∈D→x |s(z)| ≤ 1 for all x ∈ T; (iii) there exists s0 ∈ S0 and γ1, . . . , γκ ∈ D, with s0(γi) = 0, such that s(z) = κ

• i=1

1 − γiz z − γi

• s0(z);

(iv) (a) there exists an open set A of Dc such that V (s, D, A) = κ; (b) for almost all x ∈ T such that limz

→x |s(z)| exists one has that

lim

z →x |s(z)| ≤ 1.

Proof.

(i) ⇒ (ii): Trivial. (ii) ⇒ (i): A consequence of the fact that (ii)(b) implies that if V (f , D, w0) = κ0 ∈ N0 for some w0 ∈ Dc, then there exists an open non-empty neighborhood A0

• f w0 in Dc such that V (f , D, A0) = κ0. (ii) ⇒ (iv): This holds by the preceding reasoning. (iv)

⇒ (iii): Assume that (iv) holds with ∞ ∈ A (otherwise transform the function). Then there exist γ1, . . . , γκ ∈ D such that s0(z) :=

κ

• i=1

z − γi 1 − γiz s(z) is a bounded function. In light of (iv)(b), e.g. the Inner-Outer factorization (for bounded functions) implies that s0 ∈ S0. (iii) ⇒ (ii): This is obvious (with w = ∞).

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 13 / 19

A function-theoretical approach

From the kernel definition of generalized Schur functions one can deduce that generalized Schur functions have the following properties.

Lemma

Let s ∈ Sκ. Then the following statements hold: (i) V (s, D, Dc) ≤ κ; (ii) lim supz→x |s(x)| ≤ 1 for all but at most κ points x of T. Combining this properties with the preceding proposition yields us the following characterization

• f generalized Schur functions.

Theorem

Let s be meromorphic on D. Then equivalent are: (i) V (s, D, Dc) = κ; (ii) (a) there exists w ∈ Dc such that V (s, D, w) = κ; (b) lim supz∈D→x |s(z)| ≤ 1 for all x ∈ T; (iii) there exists s0 ∈ S0 and γ1, . . . , γκ ∈ D, with s0(γi) = 0, such that s(z) = κ

• i=1

1 − γiz z − γi

• s0(z);

(iv) (a) there exists an open set A of Dc such that V (s, D, A) = κ; (b) for almost all x ∈ T such that limz

→x |s(z)| exists one has that

lim

z →x |s(z)| ≤ 1;

(v) s ∈ Sκ.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 14 / 19

A function-theoretical approach

Corollary

Let f be a function meromorphic on D. Then s ∈ Sκ for some κ ∈ N0 if and only if lim supz→x |s(z)| ≤ 1 for every x ∈ T For generalized Nevanlinna functions no similar statement holds. Nevertheless, generalized Nevanlinna functions are very closely related to generalized Schur functions: f ∈ Nκ if and only if there exists s ∈ Sκ such that for z ∈ C+ f (z) = φ−1 ◦ s ◦ φ(z), φ(z) = z − i z + i . Since the preceding multivalency characterizations behave regularly with respect to this transformation, from the preceding theorem follows for instance that

Theorem

Let f be a symmetric function meromorphic on C \ R and let κ ∈ N0. Then f ∈ Nκ if and only if V (f , C+, C−) = κ. Now the crucial difference between generalized Schur and generalized Nevanlinna functions should be clear: The valency of a generalized Schur function is constant in the neighborhood of ∞ (with respect to which they are factorized), while the valency of a generalized Nevanlinna function is not constant in either a neighborhood of ∞ or 0 (with respect to which they are factorized)!

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 15 / 19

A function-theoretical approach

Accordingly, the strategy for proving the factorization property should also become clear: (1) First generalized Nevanlinna functions have to be transformed such that their valency is constant in a neighborhood of 0 and ∞ (while staying within the same class); (2) secondly, their poles and zeros should be factored out; (3) finally, a limit should be taken. In the limit process the following special case of the Arzela-Ascoli Theorem is used (see e.g. Donoghue):

Lemma

Let the infinite family F of ordinary Nevanlinna functions be uniformly bounded at a point w ∈ C+. Then there exists a subsequence in F which converges to an ordinary Nevanlinna function. Furthermore, the following characterization of ordinary Nevanlinna functions is also used:

Proposition

Let f be a symmetric function meromorphic on C \ R. Then f is an ordinary Nevanlinna function if and only if (i) there exists a non-empty open set A ⊆ C− such that V (f , C+, A) = 0; (ii) limz

→x Im f (z) ≥ 0 for almost all x ∈ R.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 16 / 19

A function-theoretical approach

Step 1: Let f ∈ Nκ be arbitrary (with κ = 0 otherwise there is nothing to prove), then ∃s ∈ Sκ such that f (z) = i 1 + s (φ(z)) 1 − s (φ(z)) , φ(z) = z − i z + i . For c ∈ R, 0 < c < 1, define sc and fc as sc(z) := c · s(z) and fc(z) := i 1 + sc (φ(z)) 1 − sc (φ(z)) = i 1 + cs (φ(z)) 1 − cs (φ(z)) . (3.1) Then sc ∈ Sκ for every c ∈ (0, 1) and V (sc, D, {z ∈ C ∪ {∞} : |z| > c}) = κ. Consequently, for every c ∈ (0, 1) fc ∈ Nκ and there exist open neighborhoods Ac

0 and Ac ∞ of 0 and ∞ in

C ∪ {∞}, respectively, such that V (fc, C+, Ac

0) = κ

and V (fc, C+, Ac

∞) = κ.

(3.2) Step 2: Let αc

1, . . . , αc κ = f −1 c

(0) ∩ C+ and βc

1, . . . , βc κ = f −1 c

(∞) ∩ C+. Consider fc,0(z) := rc(z)fc(z), rc(z) := κ

i=1(z − βc i )(z − βc i )

κ

i=1(z − αc i )(z − αc i )

. (3.3) By construction fc,0 is a bounded function on C+ with bounded inverse for which limz

→x Im f (z) ≥ 0 for almost all x ∈ R. Accordingly fc,0 is an ordinary Nevanlinna function.

Step 3: Evidently, there exists a sequence {cn}n∈N, cn ∈ (3/4, 1), converging to 1 such that αcn

i

converges to some αi ∈ C+ ∪ R ∪ {∞} and βcn

i

converges to some βi ∈ C+ ∪ R ∪ {∞}, for every 1 ≤ i ≤ κ. Both fc and f −1

c

are for c > 3/4 uniformly bounded on any nonempty compact subset K of the nonempty open set φ−1 ◦ s−1 ({z ∈ C : |z − 3| < 1}), see (3.1) and the text following it. Thus rc is uniformly bounded on K and, hence, so is fc,0. Therefore there exists by the preceding lemma a subsequence {cnk } of {cn} such that fc,0 converges to some f0 ∈ N0 as k tends to ∞. Consequently, the statement follows by taking the limit in (3.3) via the subsequence {cnk }.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 17 / 19

A function-theoretical approach

Conclusions

Multivalency is the proper tool if you want to establish factorizations; whether of generalized Schur functions or of generalized Nevanlinna functions. Multivalency allows us to define generalized Schur and Nevanlinna functions in a function-theoretic way. And, hence, also allow us to prove their factorization in a purely function-theoretical manner. Multivalency also allows us to establish the factorization of generalized Nevanlinna functions (by making use of the obtained results on generalized Schur functions) in a direct manner. Hereby also a relatively simple proof of the invariant subspace property is obtained.

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 18 / 19

A function-theoretical approach

Bibliography

• P. Delsarte, Y. Genin and Y. Kamp,

Pseudo-Carath´ eodory functions and Hermitian Toeplitz matrices Philips J. Res., 41 (1986), 1–54 V.A. Derkach, S. Hassi, and H.S.V. de Snoo Operator models associated with Kac subclasses of generalized Nevanlinna functions Methods of Funct. Anal. and Top., 5 (1999), 65–87

• A. Dijksma, H. Langer, A. Luger, and Yu. Shondin

A factorization result for generalized Nevanlinna functions of the class Nκ Integral Equations Operator Theory, 36 (2000), 121–125 M.G. Kre˘ ın and H. Langer Some propositions on analytic matrix functions related to the theory of operators Acta Sci. Math. 43 (1981), 181–205

• A. Luger

A factorization of regular generalized Nevanlinna functions

• Integr. equ. oper. theory 43 (2002), 326–345

H.L. Wietsma Generalized Nevanlinna functions and multivalency Indagationes Mathematicae 29 (2018), 997–1008 H.L. Wietsma The factorization of generalized Nevanlinna functions and the invariant subspace property Indagationes Mathematicae 30 (2019), 26–38

H.L. (Rudi) Wietsma () Factorizations, invariant subspaces and multivalency December 19, 2019 19 / 19