THE DISTANCE FROM THE APOSTOL SPECTRUM
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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 10, October 1996 THE DISTANCE FROM THE APOSTOL SPECTRUM V. KORDULA AND V. MÜLLER (Communicated by Palle E. T. Jorgensen) Abstract. If T is an s-regular operator in a Banach space (i.e. T has closed range N(T ) R (T )) γ(t ) is the Kato reduced minimum modulus, then lim γ(t n ) 1/n =supr:t λis s-regular for λ <r. Let x be an element of a Banach algebra A. The spectral radius of x is given by the well-known spectral radius formula: r(x) = lim x n 1/n. There are a number of generalizations of this formula. If we set d(x) =inf xy : y A, y =1 denote by τ l (x) =λ C:d(x λ)=0the left approximate point spectrum of x, thendist0,τ l (x) = lim d(x n ) 1/n ;see[13],[9]. In particular, in the algebra B(X) of all bounded linear operators in a Banach space X this gives formulas for radii of boundedness below or surjectivity: supr : T λ is bounded below for λ <r= lim j(t n ) 1/n supr : T λ is onto for λ <r= lim k(t n ) 1/n, where j(t )k(t) are the moduli of injectivity surjectivity of T : j(t )=inf Tx : x X, x =1 k(t )=supr:tu X ru X, where U X is the closed unit ball in X. For a bounded linear operator T in a Banach space X denote by N(T ) R(T) its kernel range, respectively. Denote further R (T )= n=1 R(T n ) N (T)= n=1 N(T n ). The injectivity surjectivity moduli of an operator which is bounded below (onto) are special cases of the Kato reduced minimum modulus [7] Tx γ(t )=inf distx, N(T ) : x X\N(T ) (for T = 0 we define formally γ(t )= ). The existence the meaning of the limit lim γ(t n ) 1/n in a more general setting were studied by Apostol [1] Mbekhta [10]. Received by the editors October 14, 1994, in revised form, January 26, Mathematics Subject Classification. Primary 47A10, 47A53. The research was supported by the grant No of the Academy of Sciences of the Czech Republic c 1996 American Mathematical Society
2 3056 V. KORDULA AND V. MÜLLER Definition. Let T B(X). We say that T is s-regular (= semi-regular) if R(T )is closed N(T ) R (T ). The s-regular operators closely related classes of operators were studied (under various names) by many authors; see [3, 4, 5, 6, 8, 16]. We list some of the most important equivalent conditions for s-regular operators; see [11, 12]. Theorem. Let T B(X) be an operator with a close range. The following conditions are equivalent: (1) T is s-regular, (2) the function λ R(T λ) is continuous at 0 in the gap topology, (3) the function λ N(T λ) is continuous at 0 in the gap topology, (4) the function λ γ(t λ) is continuous at 0, (5) lim inf λ 0 γ(t λ) > 0, (6) N (T ) R(T ), (7) N (T ) R (T ). Denote further σ γ (T )=λ C:T λis not s-regular. The set σ γ (T )was studied by Apostol [1], Rakočevič [15], Mbekhta Ouahab [11, 12] Mbekhta [10]. The terminology is not unified; we suggest to call σ γ (T ) the Apostol spectrum of T. The Apostol spectrum σ γ (T ) is always a non-empty compact subset of the complex plane, σ(t) σ γ (T) σ(t) σ γ f(t)=fσ γ (T) for any function f analytic in a neighbourhood of σ(t ). If T is an s-regular operator in a Hilbert space, then the limit lim γ(t n ) 1/n exists (1) lim γ(t n ) 1/n =dist0,σ γ (T)=supr:T λis s-regular for λ <r; see [1, Theorem 3.2] or [10, Theorem 3.1]. The aim of this paper is to prove equality (1) for operators in Banach spaces. This gives a positive answer to the conjecture of Rakočevič [15] Mbekhta generalizes the above-mentioned results for radii of injectivity surjectivity. Further, we study the essential version of this result. If T is a semi-fredholm operator, then the limit lim γ(t n ) 1/n exists by [2] it is equal to the semi-fredholm radius of T : lim γ(t n ) 1/n =supr:t λis semi-fredholm for λ <r; see [17] [2]. We prove a similar formula for essentially s-regular operators which generalizes the semi-fredholm case. The authors wish to thank M. Mbekhta for drawing their attention to the problem for fruitful discussions concerning it. Lemma 1. T B(X) is s-regular if only if there exists a closed subspace M X such that TM = M the operator T : X/M X/M induced by T is bounded below. Proof. If T is s-regular, then set M = R (T ). It is well known that M is closed (see e.g. [4, Theorem 3.4]) that TM = M T : X/M X/M is bounded from below.
3 THE DISTANCE FROM THE APOSTOL SPECTRUM 3057 Conversely, let M be the subspace of X with the required properties. Then TM = M implies M R (T ). If Tx=0,then T(x+M) = 0 the injectivity of T implies x M. ThusN(T) M R (T). It remains to prove that T has closed range. Let π : X X/M be the canonical projection. We show R(T )=π 1 R( T). If y R(T ),y = Tx for some x X, then πy = Tx+M = T(x+M) R( T)sothatR(T) π 1 R( T). If y X πy R( T), i.e. y + M = Tx+M for some x X, theny R(T)sinceM R(T). Thus R(T )=π 1 R( T), which is closed since R( T )isclosedπcontinuous. Lemma 2. Let T B(X), letmbe a closed subspace of X such that TM = M the operator T : X/M X/M induced by T is bounded below. Denote by T 1 : M M the restriction of T to M. Then lim γ(t n ) 1/n =min lim γ(t n 1 )1/n, lim γ( T n ) 1/n. Proof. The limits on the right-h side exist by [17]. If T n x =0,then T n (x+m)= 0, i.e. x M. ThusN(T n ) M N(T1 n )=N(T n ).We have γ(t1 n T n )=inf 1x distx, N(T1 n) : x M\N(T 1 n ) =inf Further, since TM = M, γ( T n )=inf Tn (x+m) x+m inf T n x distx, N(T n ) : x M\N(T n ) T n x distx, M : x M inf Thus γ(t n ) minγ(t n 1 ),γ( T n ) Denote T n x+m :x M =inf γ(t n ). distx, M : x M T n x distx, N(T n ) : x M lim sup γ(t n ) 1/n min lim γ(t n 1 ) 1/n, lim γ( T n ) 1/n. γ(t n ). s =min lim γ(t 1 n ) 1/n, lim γ( T n ) 1/n. We prove lim inf γ(t n ) 1/n s. Let n 1, x = x 0 R(T n ), x =1,lets>ε>0. Then x + M R( T n ) T i (x + M) γ( T i ) 1 x+m γ( T i ) 1 Thus there exist vectors x i T i (x + M) such that (i=1,...,n). x i γ( T i ) 1 (1 + ε) (i =1,...,n). Denote m i = Tx i+1 x i (i =0,...,). Then m i T x i+1 + x i (1 + ε) [ T γ( T i+1 ) 1 + γ( T i ) 1] (i =0,...,).
4 3058 V. KORDULA AND V. MÜLLER Further, T i (m i + M) =T i+1 x i+1 T i x i + M = M so that m i M for each i. We have T i m i =(T n x n T x )+(T x T n 2 x n 2 )+ +(Tx 1 x) = T n x n x. Since T 1 M M is onto, there exist vectors m i M such that T n i m i = m i m n i i (1 + ε)γ(t1 ) 1 m i.thus ) T (x n n = T n x n T i m i = x x n Thus m i m i (1 + ε)γ( T n ) 1 + (1 + ε) 2 γ(t1 n i ) 1[ T γ( T i+1 ) 1 + γ( T i ) 1]. γ(t n ) 1 (1 + ε)γ( T n ) 1 + (1 + ε) 2 γ(t1 n i ) 1[ T γ( T i+1 ) 1 + γ( T i ) 1]. Find n 0 such that Denote γ(t i 1 ) (s ε)i, γ( T i ) (s ε) i (i n 0 ). K = max max γ(t i 1 i n 0+1 1) 1,γ( T i ) 1,(s ε) i. For n large enough we have γ(t n ) 1 (1 + ε) 2[ n n 0 1 (s ε) n + (s ε) i n ( T (s ε) i 1 +(s ε) i ) i=n 0 n (s ε) i n( T K+K ) + K ( T (s ε) i 1 )] i +(s ε) i=n n 0 (1 + ε) 2 (s ε) n0 n[ ] K +(n 2n 0 )(K T +K)+2n 0 K( T K+K) (1 + ε) 2 (s ε) n0 n n K, where K is a constant independent of n. Hence lim inf γ(t n ) 1/n n n0 lim inf(s ε) n = s ε. Since ε>0 was arbitrary, we conclude that lim inf γ(t n ) 1/n s, sothat lim γ(t n ) 1/n = s.
5 THE DISTANCE FROM THE APOSTOL SPECTRUM 3059 Theorem 3. Let T B(X) be s-regular. Then dist0,σ γ (T)= lim γ(t n ) 1/n. Proof. Denote r = dist0,σ γ (T). Let M = R (T),T 1 = T M, let T : X/M X/M be the operator induced by T.Ifλis a complex number satisfying λ < lim γ(t n ) 1/n =minlim γ(t n 1 ) 1/n, lim γ( T n ) 1/n, then T 1 λ is onto T λ is bounded below. Thus T λ is s-regular by Lemma 1 lim γ(t n ) 1/n r. Conversely, it is well known (see e.g. [15, Theorem 5.2]) that R (T λ) is constant on the component of C\σ γ (T ) containing 0, in particular R (T λ) =M for λ <r. If λ <r,then(t λ)m=m T λ = T λ : X/M X/M is bounded below. Thus lim γ(t1 n)1/n r lim γ( T n ) 1/n r. Hence lim γ(t n ) 1/n r by Lemma 2. Remark. It is possible to deduce the inequality dist0,σ γ (T) lim γ(t n ) 1/n from [11, Theorem 2.10]. We have obtained a new direct proof of this result. Definition. T B(X) is called essentially s-regular if R(T ) is closed there exists a finite-dimensional subspace F X such that N(T ) R (T )+F. Define further σ eγ (T )=λ C:T λis not essentially s-regular. For properties of essentially s-regular operators the set σ eγ (T ) see [14, 15]. Theorem 4. Let T B(X) be essentially s-regular. Then lim γ(t n ) 1/n exists lim γ(t n ) 1/n =max r:t λis s-regular for 0 < λ <r =dist 0,σ γ (T)\0. Proof. By [14, Theorem 3.1] or [15, Theorem 2.1] there exist subspaces X 1,X 2 X such that X = X 1 X 2, dim X 1 <,TX 1 X 1,TX 2 X 2,T 1 =T X 1 is nilpotent T 2 = T X 2 is s-regular (the Kato decomposition). By the previous theorem dist0,σ γ (T 2 ) = lim γ(t2 n ) 1/n. For n dim X 1 we have T1 n =0sothat N(T n )=X 1 N(T2 n). Let P be the projection with R(P )=X 2 N(P )=X 1. Let x 2 X 2.Wehave distx 2,N(T2 n )=inf x 2 y 2 : y 2 X 2,T2 n y 2 =0 P inf y 1 (x 2 y 2 ) : y 1 X 1,y 2 X 2,T2 n y 2 =0 = P distx 2,N(T n ) P distx 2,N(T2 n ). Then γ(t2 n T n )=inf 2x 2 distx 2,N(T2 n) :x 2 X 2 \N(T2 n ) T n x 2 inf distx 2,N(T n ) :x 2 X 2 \N(T n ) =inf T n (x 1 x 2 ) distx 1 x 2,N(T n ) :x 1 x 2 X\N(T n ) =γ(t n )
6 3060 V. KORDULA AND V. MÜLLER γ(t n T2 n ) inf x 2 distx 2,N(T n ) :x 2 X 2 \N(T2 n ) P inf T n 2 x 2 distx 2,N(T n 2 ) :x 2 X 2 \N(T n 2 ) = P γ(t2 n ). Hence lim γ(t n ) 1/n = lim γ(t n 2 ) 1/n. If λ 0,thenT λis s-regular if only if T 2 λ is s-regular. Then maxr : T λ is s-regular for 0 < λ <r=dist0,σ γ (T 2 )= lim γ(t n ) 1/n. The following lemma is an analog of Lemma 1 for essentially s-regular operators: Lemma 5. T B(X) is essentially s-regular if only if there exists a closed subspace M X such that TM = M the operator T : X/M X/M induced by T is upper semi-fredholm. Proof. If T is essentially s-regular, then set M = R (T ). If X = X 1 X 2 is the Kato decomposition (dim X 1 <,TX 1 X 1,TX 2 X 2,T 1 = T X nilpotent T 2 = T X 2 s-regular), then M = R (T 2 ) X 2 TM = T 2 M = M. If x = x 1 x 2 satisfies Tx M,thenT 2 x 2 Mso that x 2 M. Thusx X 1 +M N( T ) X 1 + M. Hence dim N( T ) <. Let π : X X/M be the canonical projection. Since M R(T )R( T)= Tx+ M : x X = πr(t), the range of T is closed. Thus T is upper semi- Fredholm. Conversely, let M be a subspace of X with the required properties. We can prove that R(T ) is closed in exactly the same way as in Lemma 1. Further, M R (T ). If Tx =0,then T(x+M) = 0, i.e. πx N( T). Thus N(T ) π 1 N( T ) M +F R (T )+F for a finite-dimensional subspace F X. Theorem 6. Let T,A B(X),TA =AT, leta be a quasinilpotent. Then (1) σ γ (T + A) =σ γ (T), (2) σ γe (T + A)=σ γe (T). Proof. Let T be an essentially s-regular operator, let A be a quasinilpotent commuting with T. Denote M = R (T ),T 1 =T M,let T:X/M X/M be the operator induced by T. Clearly AM M so that we can define operators A 1 = A M Ã : X/M X/M induced by A. Clearly r(a 1) = lim A n 1 1/n lim A n 1/n =0r(Ã) = lim Ãn 1/n lim A n 1/n =0sothat σ(a 1 )=0 σ(ã) =0.FurtherT 1A 1 =A 1 T 1 T Ã = Ã T. Denote by σ δ (T )=λ C:T λis not onto, σ π (T )=λ C:T λis not bounded below, σ πe (T) =λ C:T λis not upper semi-fredholm the defect spectrum, the approximate point spectrum the essential approximate point spectrum, respectively.
7 THE DISTANCE FROM THE APOSTOL SPECTRUM 3061 By the spectral mapping property for these spectra we have σ δ (T 1 + A 1 )=σ δ (T), σ π ( T+Ã)=σ π( T), σ πe ( T + Ã) =σ πe( T). Thus 0 σ δ (T + A), i.e. (T + A)M = M. Similarly 0 σ πe ( T + Ã), i.e. T + Ã is upper semi-fredholm. By the previous lemma T + A is essentially s-regular. This proves (2). If T is s-regular A a quasinilpotent commuting with T, then in the same way (T + A)M = M T + Ã is bounded below. Hence T + A is s-regular by Lemma 1. Remark. Statement (1) for Hilbert space operators was proved in [10, Theorem 4.8]. The second statement gives a positive answer to Question 3 of [15]. References 1. C. Apostol, The reduced minimum modulus, Michican Math. J. 32 (1985), MR 87a: K.-H. Förster M.A. Kaashoek, The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator, Proc. Amer. Math. Soc. 49 (1975), MR 51: M.A. Goldman S.N. Kratchovski, On the stability of some properties of a class of linear operators, Soviet Math. Dokl. 14 (1973), ; Dokl. Akad. Nauk SSSR 209 (1973), MR 48: S. Grabiner, Uniform ascent descent of bounded operators, J. Math. Soc. Japan 34 (1982), MR 84a: M.A. Kaashoek, Stability theorems for closed linear operators, Indag. Math. 27 (1965), MR 31: T. Kato, Perturbation theory for nullity, deficiency other quantities of linear operators, J. Anal. Math. 7 (1958), MR 21: , Perturbation theory for linear operators, Springer-Verlag, Berlin, MR 34: J.P. Labrouse, Les opérateurs quasi-fredholm: une généralisation des opérateurs semi- Fredholm, Rend. Circ. Math. Palermo 29 (1989), E. Makai Jr. J. Zemánek, The surjectivity radius, packing numbers boundedness below of linear operators, Int. Eq. Oper. Th. 6 (1983), MR 84m: M. Mbekhta, Résolvant généralisé etthéorie spectrale, J. Oper. Theory 21 (1989), MR 91a: M. Mbekhta A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, vol. 22, No. XII, Pub. Irma, Lille, , Contribution àlathéorie spectrale généralisée dans les espaces de Banach, C. R. Acad. Sci. Paris 313 (1991), MR 92h: V. Müller, The inverse spectral radius formula removability of spectrum, Cas. Pest. Mat. 108 (1983), MR 85f: , On the regular spectrum, J. Operator Theory (to appear). 15. V. Rakoćevič, Generalized spectrum commuting compact perturbations, Proc. Edinb. Math. Soc. 36 (1993), MR 94g: P. Saphar, Contributions àl étude des aplications linéaires dans un espace de Banach, Bull. Soc. Math. France 92 (1964), MR 32: J. Zemánek, The stability radius of a semi-fredholm operator, Int. Eq. Oper. Th. 8 (1985), MR 86c:47014 Institute of Mathematics AV ČR, Žitná 25, Praha 1, Czech Republic address: vmuller@mbox.cesnet.cz
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