Spectral analysis of a difference operator with a growing potential

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1 Journal of Physcs: Conference Seres PAPER OPEN ACCESS Spectral analyss of a dfference operator wth a growng potental To cte ths artcle: G V Garkavenko et al 2018 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Related content - Modellng Physcs wth Mcrosoft Excel : Fast Fourer transform B V Lengme - Commutng dfference operators of rank two G S Mauleshova and A E Mronov - Rujsenaars-Macdonald-type dfference operators from Belavn model wth open boundary condtons Bo-Yu Hou, Kang-Je Sh, Yan-Shen Wang et al. Ths content was downloaded from IP address on 04/01/2019 at 19:39

2 Spectral analyss of a dfference operator wth a growng potental G V Garkavenko 1, A R Zgolch 2 and N B Uskova 3 1 Voronezh State Pedagogcal Unversty, Physco-Mathematcal Faculty, 86 Lenna St, Voronezh, Russa 2 Voronezh State Unversty, Appled Mathematcs, Informatcs and Mechancs Faculty, 1 Unverstetskaya Square, Voronezh, Russa 3 Voronezh State Techncal Unversty, Informaton Technologes and Computer Securty Faculty, 14 Moskovsky prospekt, Voronezh, Russa E-mal: arsenj112@mal.ru, nat-uskova@mal.ru Abstract. In ths paper, we study the spectral propertes of a second order dfference operator wth a growng potental. The operator acts n the complex Hlbert space l 2(Z) of square summable complex sequences ndexed by the ntegers. Ths operator s a dscrete analogue of a second order dfferental operator wth a growng complex potental. The study s based on a method of smlar operators developed by A. G. Baskakov and hs collaborators. Ths method allows us to reduce the study of the operator to one wth a block-dagonal matrx. Asymptotc estmates of egenvalues, egenvectors, and spectral projectons of a dfference operator are obtaned. 1. Introducton In ths paper, we explore the spectral propertes of dfference operators and the correspondng dscrete Sturm-Louvlle operators [1]. To ths end, we employ the method of smlar operators. The method has ts orgns n varous smlarty and perturbaton technques, such as the classcal perturbaton methods of celestal mechancs, Ljapunov s knematc smlarty method, Fredrchs method of smlar operators used n quantum mechancs, and Turner s method of smlar operators (see [2 4]). The method has been extensvely developed and used n the works of A. G. Baskakov and hs collaborators (see [2 7] and references theren). It provdes a foundaton for fndng estmates of egenvalues and egenvectors of dfference and dfferental operators wth a growng potental. Let l 2 (Z) be the complex Hlbert space of square summable sequences ndexed by the ntegers. The nner product and the norm n l 2 (Z) are defned by (x, y) = n Z x(n)y(n) and ( ) 1/2 x = x(n) 2, n Z x, y l 2 (Z), Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by Ltd 1

3 respectvely. We consder the closed lnear dfference operator E : D(E) l 2 (Z) l 2 (Z), generated by the expresson (Ex)(n) = a(n)x(n) + 2x(n) x(n 1) x(n + 1), wth the doman D(E) l 2 (Z) defned by D(E) = {x l 2 (Z), n Z x(n) 2 a(n) 2 < }. In the standard bass of the space l 2 (Z), the matrx of the operator E s gven by 1 a( 2) a( 1) a(0) a(1) a(2) The sequence a : Z C s such that lm a(n) =. We also assume that ths sequence n satsfes one of the followng two groups of condtons: 1) a() a(j) for j and d j = nf j a() a(j) as j,, j Z; 2) a() = a( ) for Z and d j as j,, j Z + = {0} N. The frst group of condtons corresponds to a growng potental n a general form, whle the second one corresponds to an even growng potental. In [1], a fnte-dmensonal analogue of the operator E was consdered and ts egenvalue estmates were obtaned by a varaton method. In ths paper, we provde estmates of egenvalues and spectral projectons of the operator E usng the method of smlar operators as n [2 4]. It s mportant to note that the method s commonly used to study the spectral propertes for dfferental operators [5 7]. It has, however, also been used for dfference operators (see [8 11]). We wrte E = A B, where A : D(A) = D(E) l 2 (Z) l 2 (Z) s defned by and B s a bounded operator gven by (Ax)(n) = a(n)x(n) + 2x(n), n Z, (Bx)(n) = x(n + 1) + x(n 1), n Z. We shall treat the operator E as the perturbaton of A by B. For a complex Hlbert space l 2 (Z) we denote by End l 2 the Banach algebra of all bounded lnear operators n l 2 (Z). The operator B belongs to End l 2 and A s a normal closed lnear operator. In the standard scheme of the method of smlar operators, the unperturbed operator s a dfferental one and the perturbaton s a multplcaton operator by a potental functon [2 7]. In ths paper, the multplcaton operator s chosen as the unperturbed one. Ths choce works because the matrx of the multplcaton operator n the standard bass of l 2 (Z) s dagonal and, therefore, the spectral propertes of ths operator are easy to obtan. The spectrum σ(a) of the operator A can be wrtten as σ(a) = l J σ l = l J{λ l }, 2

4 where J {Z, Z + }, Z + = N {0}. In the case of the frst group of condtons, λ l = a(l)+2, l Z, are smple egenvalues. The correspondng egenvectors are e l, l Z, where e l (k) = δ ek, k Z, and δ ek s the standard Kronecker delta. The spectral projectons P l = P (σ l, A) are gven by P l x = (x, e l )e l = x(l)e l, l Z, x l 2 (Z). In the other case, λ l = a(l) + 2, l N, are egenvalues of multplcty two, and λ 0 = a(0) + 2 s a smple egenvalue. The correspondng egenvectors are e l, l Z. The spectral projectons P l = P (σ l, A) are gven by P l x = (x, e l )e l + (x, e l )e l, l N, and P 0 x = (x, e l )e 0, x l 2 (Z). We wll use the notaton H k = ImP k, k J, P (m) = m P, and H (m) = ImP m. 2. Materals and methods Let H denote an abstract complex Hlbert space. We begn wth the followng defnton. Defnton 1. Two lnear operators A : D(A ) H H, = 1, 2, are called smlar, f there exsts a contnuously nvertble operator U EndH such that A 1 Ux = UA 2 x, x D(A 2 ), UD(A 2 ) = D(A 1 ). (1) The operator U s called the smlarty transform of A 1 nto A 2. Drectly from the defnton (1), we have the followng result about the spectral propertes of smlar operators. Lemma 1. Let A : D(A ) H H, = 1, 2, be two smlar operators wth the smlarty transform U. Then the followng propertes hold. (1) We have σ(a 1 ) = σ(a 2 ), σ p (A 1 ) = σ p (A 2 ), and σ c (A 1 ) = σ c (A 2 ), where σ p denotes the pont spectrum and σ c denotes the contnuous spectrum; (2) Assume that the operator A 2 admts a decomposton A 2 = A 21 A 22 wth respect to a drect sum H = H 1 H 2, where A 21 = A 2 H 1 and A 22 = A 2 H 2 are the restrctons of A 2 to the respectve subspaces. Then the operator A 1 admts a decomposton A 1 = A 11 + A 12 wth respect to a drect sum H = H 1 H 2, where A 11 = A 1 H and A 12 = A 1 H 2 are the restrctons of A 1 to the respectve nvarant subspaces. Moreover, f P s the projecton onto H 1 parallel to H 2, then P = UP U 1 s the projecton onto H 1 parallel to H 2. (3) If λ 0 s an egenvalue of the operator A 2 and x s a correspondng egenvector, then y = Ux s an egenvector of the operator A 1 correspondng to the same egenvalue λ 0. We shall need to extend Property (2) n the Lemma 1 to the case of countable drect sums. To ths end, we assume that the abstract Hlbert space H can be wrtten as H = l J H l, where each H l, l J, s a closed nonzero subspace of H, H j s orthogonal to H l for l j J, and each x H satsfes x = l J x l, where x l H l and x 2 = l J x 2 l. In other words, we have a dsjunctve resoluton of the dentty P = {P l, l J}, (2) that s a system of dempotents wth the followng propertes 1) Pl = P l, l J; 2) P j P l = δ jl P l, j, l J; 3) The seres l J P lx converges uncondtonally to x H and x 2 = l J P lx 2 ; 4) Equaltes P l x = 0, l J, mply x = 0 H; 5) H l = ImP l, x l = P l x, l J. 3

5 Defnton 2. We say that a closed lnear operator A : D(A) H H s represented as an orthogonal drect sum of bounded operators A l EndH l, l J, that s A = l J A l, f the followng three propertes hold. 1. D(A) = {x H : l J A lx l 2 <, x l = P l x, l J} and H l D(A) for all l J. 2. For each l J, the subspace H l s an nvarant subspace of the operator A and A l s the restrcton of A to H l. The operators A l, l J, are called the parts of the operator A. 3. Ax = l J A lx l, x D(A), where x l = P l x, l J, and the seres converges uncondtonally n H. Defnton 3. Gven a contnuously nvertble operator U EndH and an orthogonal decomposton of H, a U-orthogonal decomposton of H s the orthogonal drect sum H = l Z UH l. Defnton 4. Gven a contnuously nvertble operator U EndH, we say that a closed lnear operator A : D(A) H H s a U-orthogonal drect sum of bounded lnear operators à l, l J, f Ãl = UA l U 1, l J, and A = l J à l. We remark that U-orthogonal decompostons and drect sums can be vewed as orthogonal wth respect to the nner product x, y U = Ux, Uy, x, y H. An example of drect sums of operators s provded the operator A from the ntroducton. In partcular, A s an orthogonal drect sum of operators A l = A H l = (a(l) + 2)I l, where I l s the dentty operator on H l = ImP l, l J. In other words, A = l Z(a(l) + 2)I l n the frst case, and A = l Z + (a(l) + 2)I l n the second case. Ths representaton s wth respect to the orthogonal decomposton of l 2 (Z) gven by l 2 (Z) = l J H l. Consder a new resoluton of the dentty P (m) = {P (m) } {P l, l > m, l J}. Then the operator A may also be represented as an orthogonal drect sum A = A (m) = A (m) (a(l) + 2)I l, (3) l >m,l J A l l >m,l J m Z +, l J, where A (m) s the restrcton of A to H (m) = ImP (m). The representaton (3) ( ) s wth respect to the orthogonal decomposton l 2 (Z) = H (m) H l. Observe that l >m,l J A (m) = (a(l) + 2)I l wth respect to the decomposton H (m) = H j. l >m,l J l <m,l J The method of smlar operators constructs a smlarty transform for an operator A B : D(A) H H, where the spectrum of the operator A s known and has certan propertes, and the operator B s A-bounded. 4

6 Defnton 5 [2]. Let A : D(A) H H be a lnear operator. A lnear operator B : D(B) H H s A-bounded D(B) D(A) and B A = nf{c > 0 : Bx c( x + Ax ), x D(A)} <. The space L A (H) of all A-bounded lnear operators s a Banach space wth respect to the norm A. Moreover, gven λ 0 ρ(a), where ρ(a) = C \σ(a) s the resolvent set of A, we have B L A (H) f and only f B(λ 0 I A) 1 EndH and B A = B(λ 0 I A) 1 EndH defnes an equvalent norm n L A (H). The method of smlar operators uses the commutator transform ad A : D(ad A ) EndH EndH defned by ad A X = AX XA, X D(ad A ). where the doman D(ad A ) contans all X EndH such that the followng two propertes hold: 1. XD(A) D(A); 2. The operator ad A X : D(A) H (unquely) extends to a bounded operator Y EndH; we then let ad A X = Y. The key noton of the method of smlar operators s that of an admssble trplet. Defnton 6 [2]. Let M be a lnear subspace of L A (H), J : M M, and G : M B(H). The collecton (M, J, G) s called an admssble trplet for the operator A, and the space M s the space of admssble perturbatons, f the followng sx propertes hold. 1. M s a Banach space that s contnuously embedded n L A (H),.e., M has a norm such that there s a constant C > 0 that yelds X A C X for any X M. 2. J and G are bounded lnear operators; moreover, J s an dempotent. 3. (GX)D(A) D(A) and (ad A GX)x = (X JX)x, x D(A), X M; moreover GX EndH s the unque soluton of the equaton ad A Y = AY Y A = X JX, that satsfes JY = XGY, (GX)Y M for all X, Y M, and there s a constant γ > 0 such that G γ, max{ XGY, (GX)Y } γ X Y. 5. J((GX)JY ) = 0 for all X, Y M. 6. For every X M and ε > 0 there exsts a number λ ε ρ(a), such that X(A λ ε I) 1 < ε. To formulate the man theorem of the method of smlar operators [2] for an operator A B, we use the functon Φ : M M gven by Φ(X) = BGX (GX)(JB) (GX)J(BGX) + B. (4) Theorem 1 [2]. Assume that (M, J, G) s an admssble trplet for an operator A : D(A) H H and B M. Assume also that 4γ J B < 1, where γ comes from the Property 4 of Defnton 6. Then the operator A B s smlar to the operator A JX, where X M s the (unque) fxed pont of the functon Φ gven by equaton (4), and the smlarty transform of A B nto A JX s gven by I +GX EndH. Moreover, the map Φ : M M s a contracton n the ball {X M : X B 3 B }, and the fxed pont X can be found as a lmt of smple teratons: X 0 = 0, X 1 = Φ(X 0 ) = B, etc. 5

7 3. Results and dscussons. Theorem 2. There s a number k Z + and a contnuously nvertble operator U End l 2 such that operator E = A B s smlar to the operator A X 0, where X 0 End l 2, EU = U(A X 0 ) and the subspaces H (k) = ImP (k), H l = ImP l, l > k, l J, are nvarant for X 0. The operator A X 0 s the orthogonal drect sum A X 0 = A X 0(k) l >k,l J ( ) wth respect to orthogonal decomposton H = H (k) l >k,l J H j and the dmenson of H (m) s 2m + 1. Moreover, the operator E s the U-orthogonal drect sum E = U A X 0(k) l >k,l J X 0(l) X 0l U 1 ( ) wth respect to the U-orthogonal decomposton H = UH (k) l >k,l J UH l. Proof. Let M = Endl 2 L A (l 2 ) and Xx = XA 1 Ax XA 1 Ax < XA 1 ( Ax + x ), X A XA 1. We deduce Property 1 of the Defnton 6. Gven a resoluton of dentty P as n (2), t s often convenent to represent an operator X Endl 2 n terms of ts matrx. We wrte such matrces as X = (X jl ), where X jl = P j XP l, j, l J. In the case when the matrx of a lnear operator s dagonal, ths operator s the orthogonal drect sum (see defnton 2). Any operator Z Endl 2 can be expressed (see [8]) as Z = lm n p n (1 p n )Z p, Z 0 = Z j. j=p The operator Z p, p Z, s the p-th dagonal of the operator matrx of the operator Z. We defne a famly of transformers J k X as follows: J k X = P (k) XP (k) + P XP, k 0, moreover, >k, J JX = J 0 X = J P XP, where the seres J P XP s convergent, because the operator X belongs to Endl 2. Obvously, J k = 1, because J k X P (k) XP (k) P XP 2 2 X 2 2. If X = P XP (.e., the matrx of the operator X s dagonal and X s an orthogonal drect sum of operators X ), then J k X = X and J k X = X. 6

8 Let us proceed to the constructon of the operator G 0 X : Endl 2 Endl 2. Frst, we defne t on the operator blocks X j = P XP j, where X Endl 2. For each X j, j, set G 0 X j = Y j, where Y j s the soluton of the equaton AY j Y j A = X j, j,, j J, and Y = 0 for each J. Note that the last equaton can be represented n the form A Y j Y j A j = X j, (5) where A = A H, and H = RanP. Snce σ(a ) σ(a j ) = ø,, j J, t follows that equaton (5) s solvable and the followng nequalty holds: Y j c X j dst(σ, σ j ) c X j a() a(j) c(mn j( a() a(j) )) 1. In addton, set Y = 0 for each J. Now we form the operator G 0 X from the operator blocks Y j = (G 0 X) j as follows: G 0 X =,j (GX) j(see[2], [3]). The transformer G 0 X belongs to the space Endl 2 and G 0 d 1 0. In addton, we defne a famly of transformers G k X by settng It follows from the last relaton that G k X = G 0 X G 0 (Q k XQ k ) = G 0 X Q k (G 0 X)Q k, k 0. G k X cd 1 k X, γ = γ k = cd 1 k. If X, Y Endl 2, then X(G k Y ), (G k X)Y also belong to Endl 2 and X(G k Y ) γ k X Y, (G k X)Y γ k X Y. It follows from the precedng argument that condton 4 n Defnton 6 holds wth γ = γ k of the order of d 1 k. Operator J 0 s dempotent, because J 2 0 X = J 0 (J 0 X) = J 0 ( J P XP ) = j J P j ( J P XP )P j = j J P j XP j = J 0 X. It mples that condton 2 of Defnton 6 holds true. Let us check condton 3 of Defnton 6. Let Q n = <n, J P. We consder operator AQ n GX 0 A 1 and represent t as AQ n G 0 XA 1 x = Q n G 0 Xx + Q n (X J 0 X)A 1 x, x l 2 (Z). Snce Q n G 0 X G 0 Xx, Q n (X J 0 X)A 1 x (X J 0 X)A 1 x as n, then AQ n G 0 XA 1 x y 0 l 2 (Z). Let Q n G 0 XA 1 x x 0 = G 0 XA 1 x, then due to the closedness of operator A we have x 0 D(A) and Ax 0 = y 0, where y 0 = lm n y n. Condton 6 s obvous snce X(A λ ε I) 1 X (A λ ε I) 1 at that, the frst factor s fnte and the other can be chosen arbtrarly small. Thus, we have proven Lemma 2. For each k 0 the trple (Endl 2, J k, G k ) s admssble for operator A. Lemma 2, Theorem 1 and estmate (3) mply Theorem 2. 7

9 For the class of dfference operators under consderaton, one may also use as the admssble space of perturbatons the space End 1 l 2 End l 2, whch s defned as follows. An operator Y End l 2 belongs to the space End 1 l 2 f the dagonals of the matrx of Y n the standard bass of l 2 are summable. We note that the perturbaton B belongs to End 1 l 2 because the matrx of the operator B s tr-dagonal. The space End 1 l 2 was chosen as the admssble space of perturbatons n [10 11]. In ths case, an analog of Theorem 2 may be proved along the same lnes, but the operator X n the theorem must also belong to End 1 l 2. If the sequence d, J, s such that J d 2 < then the admssble space of perturbatons may be chosen as the deal S 2 (l 2 ) of Hlbert-Schmdt operators. The operator X 0 n Theorem 2 wll then also belong to the deal S 2 (l 2 ). In general, however, we do not have B S 2 (l 2 ). Ths s why we have to start wth a prelmnary smlarty transformaton of the operator A B nto the operator A B, where B S 2 (l 2 ) [7]. Lemma 3. Under the assumptons of Theorem 2 we have Proof. We have P (X B)P c 1 d 1, > k, J, P (X B BG k B)P c 2 d 2, > k, J. and X B = BG k X G k XJ k B G k XJ k (BG k X); J k (X B) = J k (BG k X) P (X B)P = P (BG k X)P d 1 Analogcally, B X 3d 1 B 2 c 1 d 1, > k, J, c 1 > 0. P (X B BG k B)P P (BG k (X B))P c 2 d 2. Assume that Theorem 2 holds true. Then the smlarty of operators A B and A J k X yelds σ(a B) = σ(a J k X 0 ) = σ(a (k) ) ( >k A ), where A (k) = (A Q k X 0 Q k ) H (k). H (l) = RanQ l and A = (P A P X 0 ) H. Snce operator X 0 s unknown and we know only the frst and the second approxmaton, then A = (P A P B P (X 0 B)) H and the frst two operators are known, whle for the thrd operator we know only estmate from Lemma 3. Analogy, A = (P A P B P (BG k (X 0 B)) H and frst three operators are known, whle for the forth operator we know only estmate from Lemma 3. The followng thorem descrbes the spectral propertes of the operator E. Theorem 3. Let the sequence a : Z C satsfy condtons 1). There s a number k Z + such that the spectrum σ(e) of the operator E satsfes σ(e) = σ (k) where σ (k) conssts of no more than 2k + 1 egenvalues. The sets σ l, l > k, are one-pont sets σ l = {µ l } and µ l = a(l) O(d 1 l ), µ l = a(l) + 2 l >k σ l, a(l + 1) 2a(l) + a(l 1) (a(l + 1) a(l))(a(l 1) a(l)) + O(d 3 l ), l Z. 8

10 The correspondng egenvectors ẽ, > k, satsfy the asymptotc estmates ẽ ỹ = O(d 2 ), > k, where ỹ l 2 (Z) and 1, = k; ỹ (k) = (a( ± 1) a()) 1, k = ± 1; 0, n other cases. The egenvectors ẽ, Z, consttute the Resz bass n the space l 2 (Z). Theorem 4. Let the sequence a : Z C satsfy condtons 2). There s a number k Z + such that the spectrum σ(e) satsfes ( ) σ(e) = σ (k) σ l, where σ (k) conssts of no more than 2k + 1 egenvalues. The sets σ l, l > k, satsfy σ l = {µ l, µ l } and µ ±l = a(l) O(d 1 l ), l > k, µ ±l = a(l) + 2 l>k a(l + 1) 2a(l) + a(l + 1) (a(l + 1) a(l))(a(l 1) a(l)) + O(d 3 l ), l > k. Let P n, n J, be the spectral projectons P n = P (σ n, E) correspondng to the sets σ n, n J, descrbed n Theorem 3 or 4. Theorem 5. We have Pn P n = O(d 1 n ), N N P n P n = O(d 1 m ), n m n m where m > k, N > k, m, N J. The spectral projectons P n also satsfy the followng estmates of unform uncondtonal equconvergence of spectral expansons: l P (σ (k), E) + P P >k <l = O(d 1 l ), for J = Z, and P (σ (k), E) + l P >k l =0 P = O(d 1 l ), for J = Z + ), where l Z +, l > k. We need the followng defnton n the monograph [12]. Defnton 7. Let C : D(C) H H be a lnear operator n the space H whose spectrum can be represented as the unon σ(c) = k J σ k (6) 9

11 of parwse dsjont sets σ k, k Z, and let P k be the Resz projecton correspondng to the spectral set σ k. An operator C s sad to be spectral wth respect to the expanson (6) (or generalzed-spectral) f the seres k P kx s convergent for any vector x H. If σ k = {λ k }, k J, are one-pont sets and CP k = λ k P k for all k, except for fntally many, then spectral wth respect to the expanson (6) operator C s spectral operator. Operator A s the spectral operator of scalar type f AP k = λ k P k for all k J. The followng asserton s a straghtforward consequence of Theorem 5. Theorem 6. The operator L s spectral operator. 4. Concluson In ths paper we studed the spectral propertes of a second order dfference operator wth a growng potental. The operator acts n the complex Hlbert space l 2 (Z) of square summable complex sequences ndexed by the ntegers. Ths operator s a dcrete analogue of a second order dfferental operator wth a growng complex potental. Asymptotc estmates of egenvalues, egenvectors and the spectral projectons were obtaned. The man method of ths exploraton was the method of smlar operators, whch allowed us to reduce the study of the operator to one wth a block-dagonal matrx. Ths method was thoroughly descrbed n the paper and ts basc defntons, such as admssble trplet and commutator transform, were gven. Man results of the paper were formuled n fve theorems and two lemmas, proofs of one theorem and two lemmas were also provded. The artcle s ten pages long and lst of references conssts of twelve ssues. Acknowledgments The research was supported by the Russan Foundaton for Basc Research (project no ). References [1] Muslmov B and Otelbaev M 1981 Comput. Math. Math. Phys [2] Baskakov A 1983 Sberan Math J [3] Baskakov A 1995 Izv. Math [4] Baskakov A, Derbushev A and Shcherbakov A 2011 Izv. Math [5] Baskakov A and Krshtal I 2013 J. Math. Anal. and Appl [6] Baskakov A, Krshtal I and Romanova E 2017 J. Evol. Equat [7] Baskakov A and Polyakov D 2017 Sberan Math. J [8] Baskakov A and Krshtal I 2014 J. Funct. Anal [9] Garkavenko G and Uskova N 2016 Proc. of Voronezh St. Unv [10] Garkavenko G, Uskova N and Zgolch A 2016 Belgorod St. Unv. Scent. Bul. Math. Phys [11] Garkavenko G and Uskova N 2017 Sberan Electron. Math. Izv [12] Dunford N and Schwartz J 1971 Lnear operators. Part III. Spectral operators (New York: Wley-Interscence) 10