PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 1, January 1998, Pages 97–103 S 0002-9939(98)04430-X
EXTENDING THE FORMULA TO CALCULATE THE SPECTRAL RADIUS OF AN OPERATOR
FERNANDO GARIBAY BONALES AND RIGOBERTO VERA MENDOZA
(Communicated by Dale Alspach)
Abstract. In a Banach space, Gelfand’s formula is used to find the spectral radius of a continuous linear operator. In this paper, we show another way to find the spectral radius of a bounded linear operator in a complete topological linear space. We also show that Gelfand’s formula holds in a more general setting if we generalize the definition of the norm for a bounded linear operator.
1. Introduction and basic definitions In all that follows E stands for a linear vector space over the field C of complex numbers. E[t] will denote a complete locally convex topological vector space, with a Hausdorff topology t,and T:E E will be a linear map. Finally, ϑ(t) will be the filter of all balanced, convex and→ closed t-neighborhoods of zero (in E). Definition 1. The linear operator T : E[t] E[t] is said to be a bounded operator, if there is a neighborhood U ϑ(t) such→ that T (U) is a bounded set. ∈ If in the definition above T (U) is a relatively compact set, then T is said to be a compact operator. Any compact operator is a bounded operator, and any bounded operator is continuous (with the t-topology) (see [5]). We recall that, given any topological linear space E[ω]andS:E[ω] E[ω]a linear operator, the resolvent of S is the set →
ρ (S)= ξ C ξI S : E[ω] E[ω] ω ∈ − → n is bijective and has a continuous inverse .
o The spectrum of S is defined by σω(S)=C ρω(S) (the set-theoretic complement in C of the resolvent set), and the spectral radius\ by sr (S)=sup λ λ σ (S) . ω | | ∈ ω n o Definition 2. Anet xα J E is said to be t-ultimately bounded (t-ub) if, { } ⊂ given any V ϑ(t) , there is a positive real number r andanindex α0 J, both depending on∈V , such that x rV α α . ∈ α ∈ ∀ ≥ 0 Received by the editors February 27, 1996. 1991 Mathematics Subject Classification. Primary 47A10; Secondary 46A03. Key words and phrases. Spectral radius, bounded linear operator, locally convex topological space, t-ultimately bounded net. Research supported by the Coordinaci´on de Investigaci´on Cient´ıfica de la UMSNH.
c 1998 American Mathematical Society
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Let us denote by Γ the set of all t-ub nets in E. Remark 1. Any bounded, convergent or Cauchy net is a t-ub net. For more details about t-ub nets we refer the reader to [1]. Γ Definition 3. Let ξ C . We will say that 1 T n t 0(Tn=T T ... T , n ∈ ξn −→ ◦ ◦ ◦ times) if, given both V ϑ(t)and xα J Γ , there exist α0 J and n0 N such that 1 T n(x ) V∈ α α {and} ∈n n . ∈ ∈ ξn α ∈ ∀ ≥ 0 ∀ ≥ 0 Γ Definition 4. γ (T ) = inf ξ 1 T n t 0 . t | | ξn −→ n o Remark 2. It is shown by Vera [4] that for a bounded operator T, we have: Γ (i) γ (T ) < , and for any ξ C such that γ (T ) < ξ , 1 T n t 0 . t ∞ ∈ t | | ξn −→ (ii) sr (T ) γ (T ), where sr (T) is the spectral radius of T . t ≤ t t (iii) When E[t] is a Banach space, γt(T )=rt(T).
In [2] it was proved, based in the above result, that γt(T )=srt(T )whenTis a compact operator. In this paper we extend that result to any bounded operator.
2. Main results From now on let T : E[t] E[t] be a bounded operator and let U ϑ(t)be such that T (U) is a bounded→ set. ∈ Let PU be the functional of Minkowski (see [3]) generated by U, which is a seminorm on E.LetE[PU] denote the vector space E with the topology given by the seminorm PU .
Remark 3. The topology on E given by the seminorm PU is coarser than the topology t (P t ). U ≤ Proposition 1. T : E[PU ] E[PU ] is a bounded operator (hence a continuous one). → Proof. Since T (U) is a bounded set and P t, T(U)isalsoa P -bounded set U ≤ U in E[PU ].
ΓP Definition 5. γ (T ) = inf ξ 1 T n U 0 . PU | | ξn −→ n o Here ΓPU convergence means that, given any net xα J E such that for { } ⊂ all α , PU (xα) r for some positive real number r ( PU -bounded net), then P ( 1 T nx ) ≤0asanetinRwhose index set is N J . U ξn α → ×
Proposition 2. γPU (T )=γt(T).
Proof. Let ξ C be such that γPU (T) < ξ , and let V ϑ(t)and xαJ Γ ∈ 1 | | ∈ { } ∈ be given. Since ξ T (U) is a bounded set, there is a positive real number r1 such 1 that T (U) V . In [1] is shown that xα J Γ r1xα J Γ. This implies r1ξ ⊂ { } ∈ ⇒{ } ∈ that there exist both α J and r > 0 such that r x r U α α , 0 ∈ 2 1 α ∈ 2 ∀ ≥ 0 i.e., PU (r1xα) r2 , that is, the net xα α α is a PU -bounded net; therefore, ≤ { } ≥ 0 α J (α α )andn Nsuch that P ( 1 T n(x )) < 1 α α ,n ∃ 1 ∈ 1 ≥ 0 1∈ U ξn α ∀ ≥ 1 ≥ n , that is, 1 T n(x ) U for those indices. Hence 1 ξn α ∈ 1 1 1 1 n+1 = ( n ) ( ) n+1 T xα T n T r1xα T U V α α1 ,n n1, ξ r1ξ ξ ∈ r1ξ ⊂ ∀ ≥ ≥
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Γ that is, 1 T n t 0 , and therefore, γ (T ) ξ. Thisimpliesthatγ (T) ξn −→ t ≤|| PU ≤ γt(T). On the other hand, let γt(T ) < ξ and xα J , a PU -bounded net; that is, x rU for all α and some r>| | 0. Then{ } 1Tx rT(U), where α ∈ {ξ α}J ⊂ ξ Γ rT(U)isat-bounded set; therefore, 1 Tx Γ . Since 1 T n t 0,given ξ { ξ α}J ∈ ξn −→ >0, α Jand n N such that 1 T n+1x = 1 T n( 1 Tx ) U ∃ 0 ∈ 0 ∈ ξn+1 α ξn ξ α ∈ α α ,n n;thatis,P ( 1 T n+1x ) for those indices. This says ∀ ≥ 0 ≥ 0 U ξn+1 α ≤ that 1 T nx is P -convergent to 0; therefore, γ (T ) ξ . This implies that ξn α U PU ≤| | γ (T ) γ (T ) . t ≤ PU Definition 6. L(E)= S : E[t] E[t] Sis a linear and continuous operator , → n o
L (E)= S L (E) S(U) is a bounded set , U ∈ n o L (E) is a vector subspace of the complex vector space L(E). U Remark 4. For the bounded operator T that we have been working on we have n n T, T ,λT,λT LU(E) for all n N and all λ C . Moreover, for any∈ S L(E), S ∈T,T S L (E∈). ∈ ◦ ◦ ∈ U Definition 7. For any operator S LU (E) , we define, taking into account that S(U) is a bounded set, the following∈ real number: S =sup P (Sx) x U . || ||U { U | ∈ } It easy to check that Sn S n S L (E)and n N. || ||U ≤|| ||U ∀ ∈ U ∀ ∈ Γ Theorem 1. If S t S in L(E), then S T S T 0 . n −→ || n ◦ − ◦ ||U → Proof. Let us prove it by way of contradiction. Let >0 be such that there exist natural numbers n1
Proposition 3. ρ (T ) ρ (T ) . t ⊂ PU Proof. Let us suppose first that γt(T ) < 1. Let ξ ρt(T) be such that ξ > 1 k ∈ | | 1 γ (T ). Then S = ∞ k+1 T is a continuous operator and S =(ξI T)− . t k=0 ξ − n 1 k Γt Set Sn = k+1PT .ThenSn S , and from Theorem 1 it follows that k=0 ξ −→ S 1 T S 1 T 0. On the other hand, S 1 T = S 1 I and S || n ◦ ξ −P ◦ ξ ||U → n ◦ ξ n+1 − ξ ◦ 1 T = S 1 I ; hence S S 0 . Thereby, given x E such that ξ − ξ || n+1 − ||U → { m}N ⊂ P (x ) 0,then P (Sx ) P [(S S )x ]+P (S x ) 0.Thisproves U m → U m ≤ U − n m U n m → that S : E[PU ] E[PU ] is a continuous operator; hence ξ ρPU (T ). Now let ξ →ρ (T ) be such that ξ γ(T). Then 1∈>1>γ(T) , which ∈ t | |≤ t |ξ| t means that 1 I T : E[P ] E[P ] is a continuous operator. Since ξI T = ξ − U → U −
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(ξ 1 )I (T 1 I),wehavethat − ξ − − ξ 1 1 1 1 1 1 1 1 1 (ξI T)− =(ξ )− I [(T I)− ( )− I] (T I)− ; − −ξ ◦ − ξ − ξ ◦ − ξ since the right hand side is the composition of three continuous operators from
E[PU ]toE[PU]wehavethatξ ρPU(T). ∈ 1 Finally, let T be such that γt(T)
This implies that γ ˆ (T ) ξ . Hence γ ˆ (T ) γP (T ). PU ≤| | PU ≤ U
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Set ξ >γˆ (T). Let xα J be a PU -bounded net in E.Then xα+N J | | PU { } { } is a Pˆ -bounded net in E/N; hence, given >0 , there are indices α J U 0 ∈ and n N such that 1 Tˆn(x + N) (U/N) α α ,n n .This 0 ∈ ξn α ∈ ∀ ≥ 0 ≥ 0 implies that for those indices 1 T nx = u + z , u U , z N ; hence ξn α α α α ∈ α ∈ P ( 1 T nx ) P (u )+P (z ) +0 = , and thus ξ >γ (T). This U ξn α ≤ U α U α ≤ | | PU implies that γ ˆ (T ) γP (T ). PU ≥ U Proposition 6. ρ (T )=ρ (Tˆ). PU PˆU
Proof. ξ ρPU (T ) ξI T : E[PU ] E[PU ] is bijective and has a continuous inverse. ∈ ⇒ − → ˆ ˆ Let us show that A :(E/N)[PU ] (E/N)[PU ] defined by A(x + N)= 1 → (ξI T)− (x)+N, which is a linear and continuous map, is the inverse function of − ξIˆ Tˆ . For this, A(ξIˆ Tˆ)(x + N)=A(ξI\T)(x + N)=A((ξI T)(x)+N)= − 1 − − − (ξI T)− (ξI T)(x)+N = x+N . In a similar way it can be proved that − − (ξIˆ Tˆ) A = I . This implies that ξ ρ (Tˆ). PˆU It− is just◦ routine to prove the set contention∈ in the other way around.
˜ Definition 11. (^E/N)[PU ] will denote the completion (as a normed space) of ˆ ˜ ˆ (E/N)[PU ], and T will denote the natural extension of T. ˜ ˆ Remark 9. (^E/N)[PU ] is a Banach space. Besides, since T is a bounded operator, T˜ : (^E/N)[P˜ ] (^E/N)[P˜ ] is a bounded operator (see [3]). U → U ˜ Remark 10. Since (^E/N)[PU ] is a Banach space we can define, as usual, the norm of T˜ ; this will be denoted by T˜ ˜ . || ||PU
Proposition 7. γ (Tˆ)=γ˜ (T˜). PˆU PU Proof. Since T˜ is an extension of Tˆ, the proof follows immediately from the defini- tions of γ (Tˆ)andγ˜ (T˜). PˆU PU
Proposition 8. ρ (Tˆ)=ρ˜ (T˜). PˆU PU ˆ ˆ ˆ ˆ Proof. If ξ ρ(T ),then ξI T :(E/N)[PU ] (E/N)[PU ] is bijective and ∈ − ˆ → ˆ 1 has a continuous inverse, so that both ξI T and (ξI T)− have a continuous − − ˜ ˜ 1 extension to (^E/N) , which are precisely ξI T and (ξI T)− respectively. − − This implies that ξ ρ(T˜). ∈ ˜ ˜ ˜ ˜ On the other hand, if ξ ρ(T ),then ξI T : (^E/N)[PU ] (^E/N)[PU ]is bijective and has a continuous∈ inverse; hence the− restrictions of those→ functions to (E/N)[Pˆ ] are precisely ξI Tˆ and its inverse function, which are continuous U − functions for being the restrictions of continuous ones. Then ξ ρ(Tˆ). ∈
Theorem 3. γt(T )=srt(T ).
Proof. By Remark 2 (ii) it suffices to show that srt(T ) γt(T ). Also, from Re- mark 2 (iii) we get ≥ (1) sr (T˜)=γ (T˜) P˜U P˜U
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˜ because (^E/N)[PU ] is a Banach space. From Propositions 2, 5 and 7 we obtain (2) γ (T )=γ (T˜) t P˜U From Propositions 3, 6 and 8 we obtain ˜ ρt(T ) ρ ˜ (T ); ⊂ PU this implies that ˜ (3) sr ˜ (T ) srt(T ). PU ≤ From (1), (2) and (3) we finally get
γ (T ) sr (T ). t ≤ t
3. A generalization of Gelfand’s formula In this part we prove that Gelfand’s formula (see [3]) applies for a bounded operator defined on a topological vector space. Following the notation from the sections above, we will show that we can use T U in Gelfand’s formula to calculate the spectral radius of T . || || ˆ Proposition 9. For any T LU (E ) , T U = T ˆ . ∈ || || || ||PU
Proof. Set r> T U .ThenT(U) rU ; hence PU (Tx) r for all x U . || ||ˆ ⊂ ˆ ≤ ∈ This implies that T ˆ r , and therefore T ˆ T U . || ||PU ≤ || ||PU ≤|| || Set r< T . Then there exists x U such that || ||U ∈ ˆ ˆ ˆ r n 1 (5) sr ˜ (T˜) = lim T˜ n . PU n →∞ || || Finally, using (4) and (5) and the corollary above, we obtain 1 n n srt(T ) = lim T . n U →∞ || || License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SPECTRAL RADIUS OF AN OPERATOR 103 References [1] C. L. DeVito, On Alaoglu’s Theorem, Bornological Spaces and the Mackey-Ulam Theorem. Math. Ann. 192, 83-89 (1972). MR 44:2014 [2] F. Garibay and R. Vera,A Formula to Calculate the Spectral Radius of a Compact Linear Operator. Internat J. Math. & Math. Sci. 20, no. 3 (1997), 585–588. [3] W. Rudin, Functional Analysis. New York, McGraw-Hill, Inc. 1991. MR 92k:46001 [4] R. Vera, Linear Operators on Locally Convex Topological Vector Spaces. Ph.D. Thesis, De- partment of Mathematics, University of Arizona, 1994. [5] K. Yosida, Functional Analysis. Springer Verlag, New York, 1965. MR 31:5054 Escuela De Ciencias F´ısico-Matematicas,´ Universidad Michoacana de San Nicolas´ de Hidalgo, Edificio B, Planta Baja, Ciudad Universitaria, Morelia, Michoacan,´ CP 58060, Mexico´ E-mail address: [email protected] E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use