On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component

For an entirely non-selfadjoint operator with spectrum at zero, the imaginary component of which has an absolutely continuous spectrum (not necessarily dissipative and having lacunas in the spectrum), triangular and functional models are constructed.

Gespeichert in:

Bibliographische Detailangaben
Datum:	2015
Hauptverfasser:	Hatamleh, R., Zolotarev, V.A.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2015
Schriftenreihe:	Журнал математической физики, анализа, геометрии
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/118025
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component / R. Hatamleh, V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 2. — С. 174-186. — Бібліогр.: 7 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-118025
record_format	dspace
spelling	irk-123456789-1180252017-05-29T03:02:29Z On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component Hatamleh, R. Zolotarev, V.A. For an entirely non-selfadjoint operator with spectrum at zero, the imaginary component of which has an absolutely continuous spectrum (not necessarily dissipative and having lacunas in the spectrum), triangular and functional models are constructed. Для вполне несамосопряженных операторов со спектром в нуле, мнимая компонента которых имеет абсолютно непрерывный спектр (не обязательно диссипативна и может иметь лакуны в спектре), построены треугольная и функциональная модели. Для цілком несамоспряжених операторів зі спектром в нулі, уявна компонента яких має абсолготно неперервний спектр (не обов'язково дисипативна та може мати лакуни в спектрі), побудовані трикутна та функціональна моделі. 2015 Article On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component / R. Hatamleh, V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 2. — С. 174-186. — Бібліогр.: 7 назв. — англ. DOI: 10.15407/mag11.02.174 1812-9471 MSC2000: 47A45 http://dspace.nbuv.gov.ua/handle/123456789/118025 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For an entirely non-selfadjoint operator with spectrum at zero, the imaginary component of which has an absolutely continuous spectrum (not necessarily dissipative and having lacunas in the spectrum), triangular and functional models are constructed.
format	Article
author	Hatamleh, R. Zolotarev, V.A.
spellingShingle	Hatamleh, R. Zolotarev, V.A. On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component Журнал математической физики, анализа, геометрии
author_facet	Hatamleh, R. Zolotarev, V.A.
author_sort	Hatamleh, R.
title	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component
title_short	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component
title_full	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component
title_fullStr	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component
title_full_unstemmed	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component
title_sort	on model representations of non-selfadjoint operators with infinitely dimensional imaginary component
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/118025
citation_txt	On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component / R. Hatamleh, V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 2. — С. 174-186. — Бібліогр.: 7 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT hatamlehr onmodelrepresentationsofnonselfadjointoperatorswithinfinitelydimensionalimaginarycomponent AT zolotarevva onmodelrepresentationsofnonselfadjointoperatorswithinfinitelydimensionalimaginarycomponent
first_indexed	2025-07-08T13:14:31Z
last_indexed	2025-07-08T13:14:31Z
_version_	1837084676052746240
fulltext	Journal of Mathematical Physics, Analysis, Geometry 2015, vol. 11, No. 2, pp. 174–186 On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component R. Hatamleh Dept. of Mathematics, Jadara University Irbid-Jordan, Jordan E-mail: raedhat@yahoo.com V.A. Zolotarev B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkiv 61103, Ukraine E-mail: vazolotarev@gmail.com Received September 9, 2013, revised December 26, 2014 For an entirely non-selfadjoint operator with spectrum at zero, the imag- inary component of which has an absolutely continuous spectrum (not neces- sarily dissipative and having lacunas in the spectrum), triangular and func- tional models are constructed. Key words: triangular model, simple spectrum, colligation, functional model. Mathematics Subject Classification 2010: 47A45. The subject of the paper is the generalization of the well-known result ob- tained by M. S. Livs̆ic for the case of infinitely dimensional imaginary component [1, 3]. Livs̆ic’s Theorem 1. Let A be an entirely non-selfadjoint [2, 3] operator in a Hilbert space H such that a) the spectrum of the operator A is concentrated at zero, σ(A) = {0}; b) A is dissipative, and its imaginary component is one-dimensional, rankAI = 1. Then the operator A is unitarily equivalent to the integration operator Ã, ( Ãf ) (x) = i ∫ l x f(t)dt, (1) acting in the space L2 [0,l], where l 6= 0 is an eigenvalue of 2AI = A−A∗ i . c© R. Hatamleh and V.A. Zolotarev, 2015 On Model Representations of Non-selfadjoint Operators... In what follows, an analogue of this theorem is obtained for the case where the imaginary component AI of the operator A is an operator with a simple absolutely continuous spectrum, also a functional model of this class of operators is constructed. The generalization of the result of Livs̆ic is based on the study of the operator of integration in the spaces of square summable functions on compacts in R2 +. It is stated that the imaginary component of the operator of integration depends on the choice of configuration of corresponding compacts. It can be non-dissipative, can have lacunas in the spectrum, can have simple spectrum, etc. The first result generalizing Livs̆ic’s Theorem 1 obtained in [7] is Theorem 5, where the compact ΩL has a triangular form and is bounded by a smooth de- creasing curve L and by the lines x = a, y = b. Since Theorems 2–5 of Sections I, II are proved in [7], they are listed here without proofs. The development of the scheme of the constructions of Sections I, II is given in Sections III–V. Thus, for a non-dissipative operator with an imaginary component having a simple ab- solutely continuous spectrum, an analogue of theorem of Livs̆ic (Theorem 7) is obtained, where the compact ΩL is represented by conjugation of two triangular domains. In Section V, Theorem 8 (an analogue of Livs̆ic’s theorem) is proved, where the imaginary component of the dissipative operator is supposed to have a lacuna in the absolutely continuous spectrum. The compact ΩL has a triangular form, and the smooth curve L possesses constant values on some inner interval. The proved Theorems 5, 7, 8 show wide possibilities for triangular models due to the choices of a compact ΩL. It turns out that the functional model for the given classes of operators can be constructed in Hilbert spaces representing symbiosis of L2 spaces and Wiener– Paley spaces of entire functions. I. Consider a continuous curve L in R2 +, L = { (x, y) : y = α(x) ∈ C1 [0,a]; α(0) = b, α(a) = 0 } , (2) generated by a smooth monotonically decreasing function α : [0, a] → C (0 < a, b < ∞). Denote by ΩL a compact in R2 + bounded by the curve L (2) and the lines x = a, y = b. Define the Hilbert space L2 ΩL formed by the complex-valued square summable functions f : ΩL → C, L2 ΩL =    f : ∫∫ ΩL \|f(x, y)\|2dxdy < ∞    . (3) Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 175 R. Hatamleh and V.A. Zolotarev Denote by Ã the linear bounded operator acting in L2 ΩL (3): ( Ãf ) (x, y) def= i a∫ x f(t, y)dt. (4) It is easy to see that 2 ( ÃIf ) (x, y) = χΩL a∫ α−1(y) f(t, y)dt, (5) where α−1(y) is a smooth monotonically decreasing function on [0, b] inverse to α(x) (α−(α(x)) = x, ∀x ∈ [0, a]), and χΩL is a characteristic function of the set ΩL. Equality (5) implies that G def= ÃIL2 ΩL = { fχΩL ∈ L2 ΩL } , (6) where f : [0, b] → C is a complex-valued function of the variable y. Specify the smooth non-negative monotonically increasing function λ(y) def= a− α−1(y) (y ∈ [0, b]). (7) Since ∫∫ ΩL \|f(y)χΩL \|2 dxdy = b∫ 0 \|f(y)\|2λ(y)dy, the subspace G (6) is isomorphic to the weighted space L2 (0,b)(λ(y)dy) def=   f : b∫ 0 \|f(y)\|2λ(y)dy < ∞    . (8) The mapping U : G → L2 (0,b)(λ(y)dy); (Uf)(y)χΩL def= f(y) (9) defines the bi-unique correspondence between G (6) and L2 (0,b)(λ(y)dy). It is obvious that U is unitary. Formula (5) yields that the operator σ̃ = U2AI \|G U∗ in the space L2 (0,b)(λ(y)dy) acts as a multiplication by the function λ(y) (7), (σ̃f) (y) def= λ(y)f(y), (10) 176 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 On Model Representations of Non-selfadjoint Operators... where f ∈ L2 (0,b)(λ(y)dy). It is easy to see that the orthoprojector PG on the subspace G (6) is given by (PGf) (x, y) def= χΩL λ(y) a∫ α−1(y) f(t, y)dt. (11) Map now the operator ϕ̃ from L2 ΩL (3) onto L2 (0,b)(λ(y)dy) (8) by the formula ϕ̃ = UPG, (12) where PG is given in (11). As a result, we obtain that the family ∆̃ = ( Ã, L2 ΩL , ϕ̃, L2 (0,b)(λ(y)dy), σ̃ ) (13) is a colligation [2, 3], where the operators Ã, ϕ̃ and σ̃ are given by (4), (12), and (10), respectively. Theorem 2. [7] The characteristic function S ∆̃ (z) [1–3] of the colligation ∆̃ (13) is a scalar operator in L2 (0,l)(λ(y)dy), (S ∆̃ (z)f)(y) = e iλ(y) z f(y), (14) where λ(y) is given by (7), and f(y) ∈ L2 (0,l)(λ(y)dy). The proof of the theorem is given in [7]. O b s e r v a t i o n 1. The operator-function S ∆̃ (z) (14) commutes with the operator σ̃ (10) for all z ∈ C (z 6= 0). II. Consider a bounded selfadjoint operator B with a simple spectrum given in a Hilbert space H. The spectrum of the operator B belongs to the interval [0, a]. Then, according to [6], the operator B is unitarily equivalent to the operator of multiplication by the independent variable ( B̂f ) (λ) = λf(λ), f(λ) ∈ L2 (0,a)(dσ(λ), (15) where σ(λ) = 〈Eλu, u〉 is a non-decreasing function on [0, a], Eλ is the resolution of identity of B, and u ∈ H is the generating vector of the operator B. Suppose that the measure dσ(λ) is absolutely continuous by the Lebesgue measure dσ(λ) = m(λ)dλ, m(λ) = σ′(λ) ≥ 0. (16) Definition 1. [7] An absolutely continuous measure dσ(λ), λ ∈ [0, a] from (16) is said to have the AC0-property if λ∫ 0 dσ(t) t < ∞ (17) Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 177 R. Hatamleh and V.A. Zolotarev for all λ ∈ [0, a]. Condition (17) requires that the improper integral converge at zero. Define the positive monotonically increasing function y(λ), y(λ) def= λ∫ 0 dσ(t) t . (18) O b s e r v a t i o n 2. ‘A priori’ we can assume that the function y(λ) maps [0, a] onto [0, b], where b is the preassigned finite positive number. If y(λ): [0, a] → [0, d] (d > 0), then, specifying the measure dσ1(λ) = b d dσ(λ), b > 0, and substituting f(λ) → √ d b f(λ) in L2 (0,a)(dσ(λ)), we obtain the Hilbert space L2 (0,a) (dσ1(λ)) isomorphic to L2 (0,a)(dσ(λ)). Besides, the function y1(λ) con- structed by dσ1(λ) (20) already possesses the values belonging to [0, b]. This procedure signifies the renormalization of the generating vector u → √ b d u since σ(λ) = 〈Eλu, u〉. Denote by λ(y) the function inverse to y(λ) (18) (y(λ(y)) = y, ∀y). Since dσ(λ) = λdy(λ), then the change of the variable λ → λ(y) transforms the space L2 (0,a)(dσ(λ)) in L2 (0,b)(λ(y)dy), in which the operator B̂ (15) acts as a multipli- cation by the function λ(y), ( B̃f ) (y) = λ(y)f(y), (19) where f(y) ∈ L2 (0,b)(λ(t)dy). Theorem 3. [7] Let B be a bounded selfadjoint operator with a simple spec- trum in H, and the spectrum of B belongs to the segment [0, a]. If the spectral measure σ(λ) of the operator B is absolutely continuous (16) and has the AC0- property (17), then the operator B is unitarily equivalent to the operator of mul- tiplication B̃ (19) by the smooth monotonically increasing function λ(y) (inverse to y(λ) (18)) in L2 (0,b)(λ(y)dy), where the finite positive number b can be chosen arbitrarily. The following statement gives the description of the commutant of the ope- rator B̂ (17). Theorem 4. [7] An arbitrary linear operator Â in L2 (0,a)(dσ(λ)) commuting with B̂ (15) is the multiplication operator ( Âf ) (λ) = a(λ)f(λ), (20) 178 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 On Model Representations of Non-selfadjoint Operators... where f(λ) ∈ L2 (0,a)(dσ(λ)), and a(λ) is a complex-valued function from L2 (0,a)(dσ(λ)), and ‖A‖ = ‖a(λ)‖L2 (0,a) (dσ(λ)). The proof of the theorem is given in [7]. The following statement generalizing the result of Livs̆ic’s Theorem 1 is true. Theorem 5. [7] Let a linear bounded dissipative completely non-selfadjoint operator A with spectrum at zero, σ(A) = {0}, be given in a Hilbert space H and (1) let the operator 2AI restricted to H1 = AIH have a simple spectrum filling the segment [0, a], 0 < a < ∞, and let its spectral function σ(λ) be absolutely continuous (16) and have the AC0-property (17); (2) for all z ∈ C, z 6= 0, let [ PH1(A− zI)−1PH1 , AI ] = 0 take place, where PH1 is an orthoprojector on H1. Then the operator A is unitarily equivalent to the integration operator Ã, ( Ãf ) (x, y) = i a∫ x f(t, y)dt, (21) in the space L2 ΩL (3), where the curve L (2) is given by the function α−1(y) = a− λ(y) and λ(y) is inverse to the function y(λ) (18). This theorem is proved in [7]. In Sections III–V we will use the constructions of Sections I, II. III. Similarly to (2), consider the curve L in R2 +, L = { (x, y) : y = α(x) ∈ C1 [0,c]; α(0) = b, α(c) = 0 } , (22) where α: [0, c] → C is a smooth monotonically decreasing function on [0, c], 0 < b, c < ∞. Let a be a point lying between 0 and c, 0 < a < c. Denote by ΩL the compact in R2 + consisting of two linked components, ΩL = Ω1 L ∪ Ω2 L. Thus, Ω1 L represents the domain bounded by the curve L and lines y = b, x = a; Ω2 L is bounded, correspondingly, by the curve L and the lines y = 0, x = a. Define the Hilbert space L2 ΩL (3) for the given ΩL. Specify in L2 ΩL the linear bounded operator ( Ãf ) (x, y) def= i a∫ x f(t, y)dt, (23) where f ∈ L2 ΩL (3). Just as in the case of the operator Ã (4), it is easy to show that for Ã (23) 2 ( ÃIf ) (x, y) = χΩL a∫ α−1(y) f(t, y)dt (24) Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 179 R. Hatamleh and V.A. Zolotarev takes place, where χΩL is the characteristic function of the set ΩL and α−1(y) is the inverse function to α(x) (α−1(α(x)) = x ∀x ∈ [0, c]). Similarly to (6), we have that G def= ÃIL2 ΩL = { fχΩL ∈ L2 ΩL } , (25) where f : [0, b] → C is a function of the variable y. Specify the smooth monoton- ically increasing function λ(y) def= a− α−1(y). (26) Obviously, this function maps the segment [0, b] on [a − c, a], where λ(d) = 0, d = y(a). Thus the function λ(y) on the segment [0, d] possesses the values from [a− c, 0] and on [d, b], correspondingly, from [0, a]. Since ∫∫ ΩL \|f(y)χΩL \|2 dxdy = b∫ d \|f(y)\|2λ(y)dy + d∫ 0 \|f(y)\|2(−λ(y))dy = b∫ 0 \|f(y)\|2\|λ(y)\|dy, the subspace G (25) is isomorphic to the weighted space L2 (0,b)(\|λ(y)\|dy) def=   f : b∫ 0 \|f(y)\|2\|λ(y)\|dy < ∞    . (27) The unitary correspondence between G (25) and L2 (0,b)(\|λ(y)\|dy) (27) is realized by the map U : G → L2 (0,b)(\|λ(y)\|dy); (Uf)(y)χΩL def= f(y). (28) The operator σ̃ = U2ÃIU ∗ in the space L2 (0,b)(\|λ(y)\|dy) acts by means of multi- plication by the function λ(y) (26), (σ̃f) (y) def= λ(y)f(y), (29) where f ∈ L2 (0,b)(\|λ(y)\|dy). It is easy to show that the orthoprojector PG on the subspace G (25) is equal to (PGf)(x, y) def= χΩL λ(y) a∫ α−1(y) f(t, y)dt. (30) 180 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 On Model Representations of Non-selfadjoint Operators... Specify the operator ϕ̃: LΩL → L2 (0,b)(\|λ(y)\|dy) by the formula ϕ̃ = UPG, (31) where PG is given by (30). Thus, the family ∆̃ = ( Ã, L2 ΩL , ϕ̃, L2 (0,b)(\|λ(y)\|dy), σ̃ ) (32) is a colligation in which the operators Ã, ϕ̃, and σ̃ are given by (23), (31), and (29), respectively. Theorem 6. The characteristic function S ∆̃ (z) of the colligation ∆̃ (32) acts in L2 (0,b)(\|λ(y)\|dy) as follows: (S ∆̃ (z)f)(y) = e iλ(y) z f(y), (33) where λ(y) is given by (26), and f ∈ L2 (0,b)(\|λ(y)\|dy). The proof of Theorem 6 is similar to that of Theorem 2 [7]. As in the previous case, it is obvious that S ∆̃ (z)σ̃ = σ̃S ∆̃ (z) for all z ∈ C (z 6= 0). IV. Analogously to the considerations of Section II, every bounded selfadjoint operator B with simple spectrum such that σ(B) ⊂ [α, β] (−∞ < α < 0 < β < ∞) is unitarily equivalent to the operator B̂ ( B̂f(λ) ) = λf(λ), (34), where f(λ) ∈ L2 (α,β)(dσ(λ)). The spectral measure dσ(λ) is supposed to be absolutely continuous (16). Definition 2. An absolutely continuous measure dσ(λ) (λ ∈ [α, β]) (16) is said to have the AC0-property if λ∫ α dσ(t) \|t\| < ∞ (35) for all λ ∈ [α, β]. Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 181 R. Hatamleh and V.A. Zolotarev Similarly to (18), define the positive monotonically increasing function y(λ) = λ∫ α dσ(t) \|t\| (36) mapping the segment [α, β] on [0, b] where 0 < b < ∞. Denote by λ(y) the function inverse to y(λ) (36). From (36) we have that λdy(λ) = dσ(λ) (λ ∈ [0, β]); −λdy(λ) = dσ(λ) (λ ∈ [α, 0]). Therefore, after changing the variable λ → λ(y), we obtain that the space L2 (α,β)(dσ(λ)) is isomorphic to the space L2 (0,b)(\|λ(y)\|dy), in which the operator B̂ acts as an operator of multiplication by the function λ(y), ( B̃f ) (y) = λ(y)f(y), (37) where f(y) ∈ L2 (0,b)(\|λ(y)\|dy). It is obvious that in this case the following analogue of Theorem 3 is true. The considerations stated above allow us to formulate the statement of The- orem 7 (similar to Theorem 6) without supposing that the initial operator A is dissipative. Theorem 7. Let A be a linear completely non-selfadjoint operator acting in a Hilbert space H with spectrum at zero, σ(A) = {0}. Then (1) the operator 2AI restricted on H1 = AIH has a simple spectrum filling the segment [a − c, a] (0 < a < c < ∞), and its spectral function σ(λ) is absolutely continuous (19) and has the AC0-property (35); (2) for all z ∈ C (z 6= 0), [ PH1(A− zI)−1PH1 , AI ] = 0 takes place, where PH1 is the orthoprojector on H1. Hence the operator A is unitarily equivalent to the operator of integration Ã, ( Ãf ) (x, y) = i a∫ x f(t, y)dt, (38) in the space L2 ΩL (3), and the curve L (22) is given by the function α−1(y) = a− λ(y), where λ(y) is the function inverse to y(λ) (36), λ(y) = y−1(λ). P r o o f. As follows from the above considerations, there exists the unitary operator U : H1 → L2 (0,b)(\|λ(y)\|dy) such that U2AI = B̃U , where B̃ is given by (37). Construct the colligation ∆ = ( A, H, UPH1 , L 2 (0,b)(\|λ(y)\|dy), B̃ ) , 182 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 On Model Representations of Non-selfadjoint Operators... where PH1 is the orthoprojector on H1. Condition (2) of the theorem yields that the characteristic function S∆(z) of the colligation ∆ commutes with the operator B̃. Applying Theorem 4, we obtain that S∆(z) is the operator of multiplication by the function exp { iz−1c(y) } in the space L2 (0,b)(\|λ(y)\|dy) in view of the known form of the characteristic function if one takes into account that σ(A) = {0}. It is easy to see that c(y) = λ(y) since lim z→∞ iz (1− S∆(z)) = B̃. Knowing λ(y), by using formula (26), we can construct the smooth decreasing function α−1(y) mapping [0, b] onto [0, c], besides, if y(0) = d, 0 < d < b, then x(d) = a. Using x(y), we specify the curve L (22) and construct the domain ΩL. Next define the colligation ∆̃ (32), where L2 ΩL is given by (3) and the operators Ã, ϕ̃, and σ̃ are given by (23), (31), and (29), respectively. To conclude the proof, it is left to note that the characteristic functions of the colligations ∆ and ∆̃ coincide in view of (33). V. Consider a piecewise decreasing curve L = { (x, y) : y = α(x) ∈ C1 [0,a3];α(0) = b2, α (a3) = 0; α(x) = b1(∀x ∈ [a1, a2])} , (39) where 0 < a1 < a2 < a3 (< ∞), 0 < b1 < b2 (< ∞), and the function α(x) decreases monotonically on the intervals [0, a1] and [a2, a3]. In the same way as in Section I, consider the compact ΩL in R2 + bounded by the curve L (42) and the lines x = a3, y = b2. Denote by L2 ΩL (3) the Hilbert space of square summable functions on the compact ΩL. In L2 ΩL , set the operator Ã by formula (4). Then, similarly to (5) (a = a3), we have 2 ( ÃIf ) (x, y) = χΩL a3∫ α−1(y) f(t, y)dt, where α−1: [0, b2] → R is a monotonically decreasing function α−1(y) = { α−1 1 (y), y ∈ [b2, b1] , α−1 2 (y), y ∈ [0, b1] , (40) where α−1 1 (α(x)) = x for all x ∈ [0, a1] and α−1 2 (α(x)) = x for all x ∈ [a2, a3]. One ought to consider ambiguity of the function α−1(y) at the point y = b1 as corresponding limits from the left for α−1 2 (y) ( from the right for α−1 1 (y)) as y → b1 − 0 (correspondingly, as y → b1 + 0). Similarly to (7), we define a non-negative piecewise non-decreasing function λ(y) = a3 − α−1(y) (y ∈ [0, b2]). (41) Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 183 R. Hatamleh and V.A. Zolotarev Using λ(y) (41), we define the space L2 (0,b)(λ(y)dy) (8) and the selfadjoint operator σ̃ (10). In this case, the spectrum of the operator σ̃ consists of two disjoint sets [0, a3 − a2] ∪ [a3 − a1, a3] and has a lacuna (a3 − a2, a3 − a1). As a result, we obtain the operator colligation ∆̃ = ( Ã, L2 ΩL , ϕ̃, L2 (0,b)(λ(y)dy), σ̃ ) , (42) where ϕ̃ and σ̃ are given by (11), (12) and (10), respectively. Theorem 2 is true. However, in representation (14) for the characteristic func- tion S ∆̃ (λ) of colligation (42), λ(y) is given by (41) and α−1(y), correspondingly, by (40). Repeating the considerations of Sections I, II, below we can get the analogue of Theorem 5. Theorem 8. Let A be a linear bounded dissipative completely non-selfadjoint operator acting in a Hilbert space H with spectrum at zero, σ(A) = 0. Then (1) the restriction of the operator 2AI on H1 = AIH is an operator with a simple spectrum filling the set [0, a3 − a2] ∪ [a3 − a1, a3], 0 < a1 < a2 < a3 < ∞, and its spectral function is absolutely continuous on this set and has the AC0- property (20); (2) for every z ∈ C, z 6= 0, [ PH1(A− zI)−1PH1 , AI ] = 0 takes place, where PH1 is an orthoprojector on H1. Then the operator A is unitarily equivalent to the operator Ã ( Ãf ) (x, y) = i a3∫ x f(t, y)dt (43) in the space L2 ΩL (3), where the curve L is given by (39) and α−1(y) by (41), λ(y) being the inverse function of y(λ) (21). VI. Let us turn to the construction of the functional model of the colligation ∆̃ (13). First, realize the Fourier transform by the variable x, F (z, y) def= Fxf(x, y) = 1√ 2π a∫ α−1(y) eizxf(x, y)dx. (44) The function F (z, y) (44) is an entire function of exponential type by the vari- able z, the adjoint indicator diagram of which coincides with the segment of the imaginary axis (iα−1(y), ia) [4]. As a result, we obtain the Hilbert space formed by the linear manifold of the functions F (z, y) satisfying the conditions: (a) F (z, y) is an entire function of exponential type by z having the adjoint diagram (iα−1(y), ia) where α−1(y) corresponds to the curve L (2); 184 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 On Model Representations of Non-selfadjoint Operators... (b) the inequality b∫ 0 dy ∫ R dz\|F (z, y)\|2 < ∞ takes place. Denote this Hilbert space by W 2 L. The Wiener–Paley and Plancherel Theorems [4, 5] imply that the transform Fx (47) defines the unitary isomorphism between the spaces L2 ΩL (3) and W 2 L. Calculate the Fourier transform (44) of the operator Ã (4), Fx ( Ãf ) (x, y) = 1√ 2π a∫ α−1(y) eizxi a∫ x f(t, y)dtdx = i√ 2π a∫ α−1(y) dtf(t, y) t∫ α−1(y) eizxdx = i√ 2π a∫ α−1(y) dtf(t, y) eizt − eizα−1(y) iz = 1 z { F (z, y)− eizα−1(y)F (0, y) } . Thus, the operator Ã (4) in the space W 2 L acts as follows: ( ÂF ) (z, y) def= 1 z { F (z, y)− eizα−1(y)F (0, y) } , (45) where F ∈ W 2 L. The Fourier transform (44) maps the space G (6) onto the space G̃ def= { eiza − eizα−1(y) iz f(y) : f ∈ L2 (0,b)(λ(y)dy) } . (46) Taking into account (11) and (44), we obtain that the orthoprojector P G̃ in the space W 2 L on G̃ is given by ( P G̃ F ) (z, y) def= √ 2π eiza − eizα−1(y) izλ(y) F (0, y). (47) The operator ϕ̂, which is unitarily equivalent to ϕ̃ (12), is equal to (ϕ̂F )(z, y) def= √ 2π F (0, y) λ(y) . (48) It is obvious that the adjoint operator ϕ̂∗ acting from L2 (0,b) (8) into W 2 L is given by (ϕ̂∗f)(y) = √ 2π eiza − eizx(y) iz f(y). (49) Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2 185 R. Hatamleh and V.A. Zolotarev Thus, we obtain the colligation ∆̂ = ( Â,W 2 L, ϕ̂, L2 (0,b)(λ(y)dy), σ̃ ) , (50) where the spaces W 2 L and L2 (0,b)(λ(y)dy) are defined above and the operators Â, ϕ̂, and σ̂ are given by (45), (48), and (10), respectively. Theorem 9. Let A be a linear completely non-selfadjoint dissipative operator acting in a Hilbert space H, the spectrum of which is concentrated at zero, σ(A) = {0}. Then (1) the operator 2AI restricted on H1 = AIH has a simple spectrum filling the segment [0, a], 0 < a < ∞, and its spectral function σ(λ) is absolutely continuous and has the AC0-property (20); (2) for all z ∈ C, z 6= 0, [ PH1(A− zI)−1PH1 , AI ] = 0 takes place, where PH1 is the orthoprojector on H1. Then the operator A is unitarily equivalent to the operator Â ( ÂF ) (z, y) = 1 z { F (z, y)− eizα−1(y)F (0, y) } in the space W 2 L, where the curve L (2) is given by the function α−1(y), λ(y) is the function inverse to y(λ) from (21). Theorem 9 can be proved for the case of the domain ΩL considered in Sections III, V. References [1] M.S. Livs̆ic, On Spectral Decomposition of Linear Non-Selfadjoint Operators. — Matem. sb. 34/76 (1954), No. 1, 145–198. (Russian) [2] M.S. Livs̆ic and A.A. Yantsevich, Theory of Operator Colligations in Hilbert Spaces. Kharkov University Publishing House, Kharkov, 1971. (Russian) [3] V.A. Zolotarev, Analitic Methods of Spectral Representations of Non-Selfadjoint and Nonunitary Operators. Kharkov University Publishing House, Kharkov, 2003. (Russian) [4] B.Ya. Levin, Lectures on Entire Functions. Amer. Math. Soc. Providence, RI, 150, 1996. [5] N.I. Akhiezer, Lectures on Integral Transforms. Kharkov University Publishing House, Kharkov, 1984. (Russian) [6] N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space. Vol. 1, Vol. 2. Kharkov University Publishing House, Kharkov, 1978. (Russian) [7] R. Hatamleh and V.A. Zolotarev, On Many-Dimensional Model Representations of One Class of Commuting Operators. — Ukr. Math. Jornal 66 (2014), No. 1, 108–127. 186 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 2

On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component

Institution

Ähnliche Einträge