Skew exactness perturbation Robin Harte and David Larson The first author was partially supported by Enterprise Ireland grant number IC/2001/027 Abstract We offer a perturbation theory for finite ascent and descent properties of bounded operators. There are various degrees of “skew exactness” ([10];[7] (10.9.0.1), (10.9.0.2)) between compatible pairs of operators, bounded and linear between normed spaces: 1. Definition Suppose T : X → Y and S : Y → Z are bounded and linear between normed spaces; then we may classify the pair (S, T ) as left skew exact if there is inclusion S −1 (0) ∩ T (X) = {0} ,

1.1

strongly left skew exact if there is k > 0 for which 1.2

kT (·)k ≤ kkST (·)k ,

and splitting left skew exact if there is R ∈ BL(Z, Y ) for which 1.3

T = RST .

Also we may classify the pair (S, T ) as right skew exact if there is inclusion S −1 (0) + T (X) = Y ,

1.4

strongly right skew exact if there is k > 0 for which: for every y ∈ Y there is x ∈ X for which 1.5

Sy = ST x with kxk ≤ kkyk ,

and splitting right skew exact if there is R ∈ BL(Y, X) for which 1.6

S = ST R .

It is easy to see that 2. Theorem In the notation of Definition 1, there is implication 2.1

(1.3) =⇒ (1.2) =⇒ (1.1)

and 2.2

(1.6) =⇒ (1.5) =⇒ (1.4) .

Proof. Most of this holds slightly more generally ([7] Theorems 10.1.2, 10.1.4), with a general operator R′ : X → Z in place of the product ST . Note that (cf [3] (6.1)) (1.1) holds iff (ST )−1 (0) ⊆ T −1 (0) ,

2.3 and that (1.4) holds iff 2.4

S(Y ) ⊆ ST (X) • For Hilbert spaces X, Y, Z there is ([7] Theorem 10.8.1) implication (1.2)=⇒(1.3) and (1.5)=⇒(1.6).

1

A slightly stronger version of the condition (1.1) asks that S −1 (0) ∩ cl T (X) = {0} ,

2.5

which says that the operator KM JN is one one, where (cf Yang [11];[5]) KM : Y → Y /M and JN : N → Y are the natural quotient and injection induced by the subspaces M = cl T X and N = S −1 (0). Stronger again is the condition that there be k > 0 for which there is implication 2.6

y ∈ S −1 (0) =⇒ kyk ≤ k dist(y, T (X)) ,

which says that the same operator KM JN is bounded below. Evidently 2.7

(1.2) =⇒ (2.6) =⇒ (2.5) =⇒ (1.1) :

if k > 0 satisfies (1.2) and if Sy = 0 then kyk ≤ ky − T xk + kT xk ≤ ky − T xk + kkS(T x − y)k ≤ (1 + kkSk)ky − T xk • Condition (2.6), with k = 1, has been noticed by Anderson [1], who describes it by calling T (X) orthogonal to S −1 (0). Turnsek [13] has observed that it holds for certain operators on Banach algebras: 3. Theorem If S ∈ BL(Y, Y ) then (2.6) holds with k = 1 for (S, S) provided 3.1

kI − Sk ≤ 1 .

Proof. Following the argument of Turnsek ([13] Theorem 1.1) write S = I − U and Vn = I + U + . . . + U n , so that SVn = I − U n+1 = Vn S

3.2 and we have

Sy = 0 =⇒ (n + 1)y = Vn y = (I − U n+1 )x + Vn (y − Sx) and hence kyk ≤

2 kxk + ky − Sxk ; n+1

now let n → ∞ • The argument of Theorem 3 suggests - wrongly - that we are using a weakened version of the condition (1.3): we call the pair (S, T ) almost left skew exact if there are (Rn ) in BL(Z, Y ) with 3.3

kT − Rn ST k → 0 and supn kRn k < ∞ ,

and almost right skew exact if instead (Rn ) in BL(Y, X) with 3.4

kS − ST Rn k → 0 and supn kRn k < ∞ .

Also call (S, T ) almost strongly right skew exact if there is k > 0 for which: for every y ∈ Y there is (xn ) in X for which 3.5

kSy − ST xn k → 0 with supn kxn k ≤ kkyk .

Evidently (cf [10] Theorem 10.1.2) 3.6

(1.3) =⇒ (3.3) =⇒ (1.2)

and 3.7

(1.6) =⇒ (3.4) =⇒ (3.5) ;

thus (3.3) implies (2.6). We do not however derive (3.3) for (S, S) from the condition (3.1). We also remark that, whenever the space Z is complete, there is implication 3.8

(1.4) =⇒ (3.5) :

this ([2];[4] Theorem 1.1; [7] Theorem 10.5.5) uses Baire’s theorem. 2

Under certain circumstances the “left” and “right” skew exactnesses are equivalent; we begin (cf [3] Lemma 6.2) by extending the finite ascent/descent characterizations: 4. Theorem Suppose, under the conditions of Definition 1, that W ⊆ X with T (W ) ⊆ S −1 (0), and that V ⊆ Y with T (X) ⊆ S −1 (V ). Then each of the following conditions is equivalent to (1.1): 4.1

T ∨ : X/T −1(0) → Y /S −1 (0) one one ;

4.2

S ∧ : T (X) → V is one one .

Also each of the following conditions is equivalent to the condition (1.4): 4.3

S ∧ : T (X) → S(Y ) onto ;

4.4

T ∨ : X/W → Y /S −1 (0) is onto .

Proof. The equivalences (1.1)⇐⇒(4.1) and (1.4)⇐⇒(4.3) are clear. We claim that (1.1) is equivalent to (4.2) with V = Z, and that this in turn is equivalent to (4.2) for arbitrary V for which T (X) ⊆ S −1 V . The second equivalence is clear; for the first note that for arbitrary x ∈ X there is implication S(T x) ∈ S −1 (0) ⇐⇒ ST x = 0 . We also claim that (1.4) is equivalent to (4.4) with W = {0}, and that this in turn is equivalent to (4.4) for arbitrary W for which T (W ) ⊆ S −1 (0). The second equivalence is clear; for the first note that for arbitrary y ∈ Y there is implication y ∈ S −1 (0) + T (X) ⇐⇒ Sy ∈ S(T X) • If in particular X = Y = Z and ST = T S then (4.2) applies with V = T (X), and (4.4) applies with W = S −1 (0). We apply this in particular with S = T k for some k ∈ N: 5. Theorem If X = Y = Z and S = T k : Y → Y , with T in the “commutative closure” of the invertibles, in the sense that there are (Rn ) in BL(X, X) with 5.1

Rn ∈ BL−1 (X, X) ; Rn T = T Rn ; kRn − T k → 0 ,

then the following are equivalent: 5.2

(ST )−1 (0) ⊆ T −1 (0) and T (X) = cl T (X) ;

5.3

S(Y ) ⊆ ST (X) and T (X) = cl T (X) .

Proof. We recall ([5];[7] Theorem 3.5.1) that for bounded linear operators T : X → Y between (possibly incomplete) normed spaces 5.4

T bounded below and a limit of dense =⇒ T almost open ,

and hence ([5];[7] Theorem 5.5.6) by duality 5.5

T almost open and a limit of bounded below =⇒ T bounded below .

Now if Rn commutes with T then it leaves both T (X) and S −1 (0) invariant, and if Rn is invertible then (cf [7] Theorem 3.11.1) its restriction Rn∧ to T (X) will be bounded below and its quotient on Y /S −1 (0) will be onto. Thus if we assume (5.2) then by (4.1) and closed range T ∨ will be bounded below and the limit of onto Rn∨ , therefore onto, giving (5.3). If instead we assume (5.3) then by (4.3) S ∧ will be onto and by closed range almost open, and the limit of bounded below (Rnk )∧ , therefore bounded below, giving (5.2) • 3

(5.2) and (5.3) are together equivalent to the condition that T ∈ BL(X, X) is polar ([7] Definition 7.5.2), in the sense that 0 ∈ C is at worst a pole of the resolvent function (zI − T )−1 . If we relax the closed range condition we can still [12] get one of the implications, provided we further tighten the approximation by commuting invertible operators: 6. Theorem Suppose that S = T k and that 0 6∈ int σ(T ). If the finite descent condition (1.4) holds then so also does the finite ascent condition (5.2), including closed range. Proof. This is shown on Hilbert space ([12] Lemma 2.5) by Herrero, Larson and Wogen. Alternatively, since we are assuming that 0 is at worst on the boundary of the spectrum then we can take the approximating invertible operators Rn = T − λn I to be scalar perturbations of the operator T . Now if (1.4) holds, then the quotient operator T ∨ on X/S −1 (0) is (4.5) onto, and the limit of operators (T − λn I)∨ , which we claim are invertible. As in Theorem 5 it is clear that the quotient (T − λn I)∨ is onto: we claim it is also one one. To see this recall that the operator T − λn I is one-one and the restriction (T − λn I)∧ = (−λn I)∧ to the subspace T −1 (0) is onto, so that ([4] Theorem 3.11.2) the induced quotient is also one one. For the closed range note that T (X) now has a closed complement, and appeal to the “Lemma of Neuberger” ([7] Theorem 4.8.2) • Theorem 6 does not reverse: 7. Example If 7.1

S = I − λU or S = I − λV or S = λW ,

where |λ| = 1, U and V are the forward and backward shifts on ℓ2 , and W the standard weight, 7.2

(U x)1 = 0 , (U x)n+1 = xn ; (V x)n = xn+1 ; (W x)n = (1/n)xn ,

then S is one one and not onto, therefore of finite descent and not of finite ascent, while 7.3

kI − Sk = 1 so that 0 6∈ int σ(S) .

Proof. This is easily checked: note that, extended to all sequences, there is equivalence, for arbitrary x ∈ X N , 7.4

x ∈ (I − λU )−1 ⇐⇒ x ∈ (I − λV )−1 ⇐⇒ x = x1 (1, λ, λ2 , . . .) •

We need some auxiliary subspaces: 8. Definition If T ∈ BL(X, X) write 8.1

T −∞ (0) =

∞ [

T −n (0) and T ∞ (X) =

n=1

∞ \

T n (X)

n=1

for the hyperkernel and the hyperrange of T , and X \ 8.2 EX (T ) = (T − λI)−∞ (0) and FX (T ) = (T − λI)∞ (X) . λ∈C

λ∈C

Each of the subspaces in Definition 8 is linear, not necessarily closed, and hyperinvariant under T . We recall that T ∈ BL(X, X) is called algebraic if there is a nontrivial polynomial 0 6= p ∈ Poly for which 8.3

p(T ) = 0 ;

more generally T is said to be locally algebraic if [ 8.4 X = {p(T )−1 (0) : 0 6= p ∈ Poly} . For the record 4

9. Theorem If T ∈ BL(X, X) for a Banach space X then 9.1

T locally algebraic =⇒ T algebraic .

Necessary and sufficient for T to have finite descent is that 9.2

EX (T ) + T (X) = X .

Proof. The first part of this is known as Kaplansky’s Lemma; the proof [9] is a combination of Baire’s theorem and the Euclidean algorithm for polynomials. The Euclidean algorithm also gives equality [ 9.3 EX (T ) = {p(T )−1 (0) : 0 6= p ∈ Poly} = {x ∈ X : dim Poly(T )x < ∞} , and dually 9.4

FX (T ) =

\ {p(T )(X) : 0 6= p ∈ Poly} .

Then again with a combination of Baire’s theorem and the Euclidean algorithm, if T ∈ BL(X, X) there is ([12] Lemma 2.4) k ∈ N for which EX (T ) + T (X) = T −∞ (0) + T (X) = T −k (0) + T (X) •

9.5

Dually, using the Euclidean algorithm, we get half way: FX (T ) ∩ T −1 (0) = T ∞ (X) ∩ T −1 (0) .

9.6

For the essence of a possible spectral mapping theorem (cf [10]), we have 10. Theorem If S, T ∈ BL(X, X) satisfy ST = T S and either S ∈ {T k : k ∈ N}

10.1 or 10.2

V S − T U = I with {U, V } ⊆ comm(S, T ) ,

then there is equivalence 10.3

ST of finite ascent ⇐⇒ S , T of finite ascent ,

and equivalence 10.4

ST of finite descent ⇐⇒ S , T of finite descent .

Proof. The backward implications are easy ([7] Theorem 7.9.2): if S and T commute and satisfy S −k (0) = S −k−1 (0) and T −k (0) = T −k−1 (0) then (ST )−k (0) = S −k T −k (0) = S −k T −k−1 (0) = T −k−1 S −k (0) = T −k−1 S −k−1 (0) = (ST )−k−1 (0) , if instead ST = T S with S k X = S k+1 X and T k X = T k+1 X then (ST )k X = S k T k (X) = S k T k+1 X = T k+1 S k X = T k+1 S k+1 X = (ST )k+1 X . Also the forward implications are clear when (10.1) S = T k is a power of T ; if instead we assume (10.2) then we argue (ST )−1 (0) ⊆ T −1 (0) + T (X) and (ST )X ⊇ T −1 (0) ∩ T (X) ,

10.5

while if (U, V ) satisfies (10.2) then for arbitrary k ∈ N Vk S k − T k Uk = I with {Uk , Vk } ⊆ comm(S k , T k ) .

10.6 To verify (10.5) argue

ST x = 0 =⇒ x + T U x = V Sx ∈ T −1 (0) ; T (T x) = 0 =⇒ T x = T V Sx − T U T x = (ST )(V x) . For (10.6) note that for arbitrary k ∈ N V S − T U = I =⇒ V k+1 S k+1 − T U (I + V S + . . . + V k S k ) = I • 5

For an induced “spectrum” to be a closed set we have 11. Theorem T ∈ BL(X, X) is of finite descent then so is T − λI for sufficiently small λ ∈ C. Proof. This has been shown on Hilbert space by Han/Larson/Pan ([11] Lemma 2.2, Theorem 2.4). It is clear from the open mapping theorem (applied to the condition (4.4) with W = {0}) that if the condition (1.4) holds then also S −1 (0) + (T − U )(X) = Y whenever T − U ∈ BL(X, Y ) is sufficiently close to T ∈ BL(X, Y ): the problem is that we must also perturb S. However if S = T k and U = λI, so that EX (T − U ) = EX (T ), then we can argue EX (T − U ) + (T − U )(X) = EX (T ) + (T − U )(X) ⊇ S −1 (0) + (T − U )(X) = X • The subspaces of Definition 8 lead to certain special kinds of operator: 12. Definition We shall call T ∈ BL(X, X) triangular if the subspace EX (T ) is dense: 12.1

cl EX (T ) = X .

Dually T ∈ BL(X, X) is co-triangular if the subspace FX (T ) is trivial: 12.2

FX (T ) = {0} .

The shifts of Example 7 are either triangular or co-triangular: 13. Example On each of the spaces c0 and ℓp (1 ≤ p < ∞), the forward shift U is triangular, the backward shift V is co-triangular and the standard weight W is both triangular and co triangular. Proof. The hyperkernel of the backward shift is dense, since it includes all the “terminating” sequences: V −∞ (0) ⊇ c00 .

13.1 Thus

E(V ) ⊇ V −∞ (0) is dense

13.2 and also 13.3

F (V ) =

\

X

(V − λI)∞ (X) ⊇

(V − λI)−∞ (0) ⊇ V −∞ (0) is dense .

|λ|

1.1

strongly left skew exact if there is k > 0 for which 1.2

kT (·)k ≤ kkST (·)k ,

and splitting left skew exact if there is R ∈ BL(Z, Y ) for which 1.3

T = RST .

Also we may classify the pair (S, T ) as right skew exact if there is inclusion S −1 (0) + T (X) = Y ,

1.4

strongly right skew exact if there is k > 0 for which: for every y ∈ Y there is x ∈ X for which 1.5

Sy = ST x with kxk ≤ kkyk ,

and splitting right skew exact if there is R ∈ BL(Y, X) for which 1.6

S = ST R .

It is easy to see that 2. Theorem In the notation of Definition 1, there is implication 2.1

(1.3) =⇒ (1.2) =⇒ (1.1)

and 2.2

(1.6) =⇒ (1.5) =⇒ (1.4) .

Proof. Most of this holds slightly more generally ([7] Theorems 10.1.2, 10.1.4), with a general operator R′ : X → Z in place of the product ST . Note that (cf [3] (6.1)) (1.1) holds iff (ST )−1 (0) ⊆ T −1 (0) ,

2.3 and that (1.4) holds iff 2.4

S(Y ) ⊆ ST (X) • For Hilbert spaces X, Y, Z there is ([7] Theorem 10.8.1) implication (1.2)=⇒(1.3) and (1.5)=⇒(1.6).

1

A slightly stronger version of the condition (1.1) asks that S −1 (0) ∩ cl T (X) = {0} ,

2.5

which says that the operator KM JN is one one, where (cf Yang [11];[5]) KM : Y → Y /M and JN : N → Y are the natural quotient and injection induced by the subspaces M = cl T X and N = S −1 (0). Stronger again is the condition that there be k > 0 for which there is implication 2.6

y ∈ S −1 (0) =⇒ kyk ≤ k dist(y, T (X)) ,

which says that the same operator KM JN is bounded below. Evidently 2.7

(1.2) =⇒ (2.6) =⇒ (2.5) =⇒ (1.1) :

if k > 0 satisfies (1.2) and if Sy = 0 then kyk ≤ ky − T xk + kT xk ≤ ky − T xk + kkS(T x − y)k ≤ (1 + kkSk)ky − T xk • Condition (2.6), with k = 1, has been noticed by Anderson [1], who describes it by calling T (X) orthogonal to S −1 (0). Turnsek [13] has observed that it holds for certain operators on Banach algebras: 3. Theorem If S ∈ BL(Y, Y ) then (2.6) holds with k = 1 for (S, S) provided 3.1

kI − Sk ≤ 1 .

Proof. Following the argument of Turnsek ([13] Theorem 1.1) write S = I − U and Vn = I + U + . . . + U n , so that SVn = I − U n+1 = Vn S

3.2 and we have

Sy = 0 =⇒ (n + 1)y = Vn y = (I − U n+1 )x + Vn (y − Sx) and hence kyk ≤

2 kxk + ky − Sxk ; n+1

now let n → ∞ • The argument of Theorem 3 suggests - wrongly - that we are using a weakened version of the condition (1.3): we call the pair (S, T ) almost left skew exact if there are (Rn ) in BL(Z, Y ) with 3.3

kT − Rn ST k → 0 and supn kRn k < ∞ ,

and almost right skew exact if instead (Rn ) in BL(Y, X) with 3.4

kS − ST Rn k → 0 and supn kRn k < ∞ .

Also call (S, T ) almost strongly right skew exact if there is k > 0 for which: for every y ∈ Y there is (xn ) in X for which 3.5

kSy − ST xn k → 0 with supn kxn k ≤ kkyk .

Evidently (cf [10] Theorem 10.1.2) 3.6

(1.3) =⇒ (3.3) =⇒ (1.2)

and 3.7

(1.6) =⇒ (3.4) =⇒ (3.5) ;

thus (3.3) implies (2.6). We do not however derive (3.3) for (S, S) from the condition (3.1). We also remark that, whenever the space Z is complete, there is implication 3.8

(1.4) =⇒ (3.5) :

this ([2];[4] Theorem 1.1; [7] Theorem 10.5.5) uses Baire’s theorem. 2

Under certain circumstances the “left” and “right” skew exactnesses are equivalent; we begin (cf [3] Lemma 6.2) by extending the finite ascent/descent characterizations: 4. Theorem Suppose, under the conditions of Definition 1, that W ⊆ X with T (W ) ⊆ S −1 (0), and that V ⊆ Y with T (X) ⊆ S −1 (V ). Then each of the following conditions is equivalent to (1.1): 4.1

T ∨ : X/T −1(0) → Y /S −1 (0) one one ;

4.2

S ∧ : T (X) → V is one one .

Also each of the following conditions is equivalent to the condition (1.4): 4.3

S ∧ : T (X) → S(Y ) onto ;

4.4

T ∨ : X/W → Y /S −1 (0) is onto .

Proof. The equivalences (1.1)⇐⇒(4.1) and (1.4)⇐⇒(4.3) are clear. We claim that (1.1) is equivalent to (4.2) with V = Z, and that this in turn is equivalent to (4.2) for arbitrary V for which T (X) ⊆ S −1 V . The second equivalence is clear; for the first note that for arbitrary x ∈ X there is implication S(T x) ∈ S −1 (0) ⇐⇒ ST x = 0 . We also claim that (1.4) is equivalent to (4.4) with W = {0}, and that this in turn is equivalent to (4.4) for arbitrary W for which T (W ) ⊆ S −1 (0). The second equivalence is clear; for the first note that for arbitrary y ∈ Y there is implication y ∈ S −1 (0) + T (X) ⇐⇒ Sy ∈ S(T X) • If in particular X = Y = Z and ST = T S then (4.2) applies with V = T (X), and (4.4) applies with W = S −1 (0). We apply this in particular with S = T k for some k ∈ N: 5. Theorem If X = Y = Z and S = T k : Y → Y , with T in the “commutative closure” of the invertibles, in the sense that there are (Rn ) in BL(X, X) with 5.1

Rn ∈ BL−1 (X, X) ; Rn T = T Rn ; kRn − T k → 0 ,

then the following are equivalent: 5.2

(ST )−1 (0) ⊆ T −1 (0) and T (X) = cl T (X) ;

5.3

S(Y ) ⊆ ST (X) and T (X) = cl T (X) .

Proof. We recall ([5];[7] Theorem 3.5.1) that for bounded linear operators T : X → Y between (possibly incomplete) normed spaces 5.4

T bounded below and a limit of dense =⇒ T almost open ,

and hence ([5];[7] Theorem 5.5.6) by duality 5.5

T almost open and a limit of bounded below =⇒ T bounded below .

Now if Rn commutes with T then it leaves both T (X) and S −1 (0) invariant, and if Rn is invertible then (cf [7] Theorem 3.11.1) its restriction Rn∧ to T (X) will be bounded below and its quotient on Y /S −1 (0) will be onto. Thus if we assume (5.2) then by (4.1) and closed range T ∨ will be bounded below and the limit of onto Rn∨ , therefore onto, giving (5.3). If instead we assume (5.3) then by (4.3) S ∧ will be onto and by closed range almost open, and the limit of bounded below (Rnk )∧ , therefore bounded below, giving (5.2) • 3

(5.2) and (5.3) are together equivalent to the condition that T ∈ BL(X, X) is polar ([7] Definition 7.5.2), in the sense that 0 ∈ C is at worst a pole of the resolvent function (zI − T )−1 . If we relax the closed range condition we can still [12] get one of the implications, provided we further tighten the approximation by commuting invertible operators: 6. Theorem Suppose that S = T k and that 0 6∈ int σ(T ). If the finite descent condition (1.4) holds then so also does the finite ascent condition (5.2), including closed range. Proof. This is shown on Hilbert space ([12] Lemma 2.5) by Herrero, Larson and Wogen. Alternatively, since we are assuming that 0 is at worst on the boundary of the spectrum then we can take the approximating invertible operators Rn = T − λn I to be scalar perturbations of the operator T . Now if (1.4) holds, then the quotient operator T ∨ on X/S −1 (0) is (4.5) onto, and the limit of operators (T − λn I)∨ , which we claim are invertible. As in Theorem 5 it is clear that the quotient (T − λn I)∨ is onto: we claim it is also one one. To see this recall that the operator T − λn I is one-one and the restriction (T − λn I)∧ = (−λn I)∧ to the subspace T −1 (0) is onto, so that ([4] Theorem 3.11.2) the induced quotient is also one one. For the closed range note that T (X) now has a closed complement, and appeal to the “Lemma of Neuberger” ([7] Theorem 4.8.2) • Theorem 6 does not reverse: 7. Example If 7.1

S = I − λU or S = I − λV or S = λW ,

where |λ| = 1, U and V are the forward and backward shifts on ℓ2 , and W the standard weight, 7.2

(U x)1 = 0 , (U x)n+1 = xn ; (V x)n = xn+1 ; (W x)n = (1/n)xn ,

then S is one one and not onto, therefore of finite descent and not of finite ascent, while 7.3

kI − Sk = 1 so that 0 6∈ int σ(S) .

Proof. This is easily checked: note that, extended to all sequences, there is equivalence, for arbitrary x ∈ X N , 7.4

x ∈ (I − λU )−1 ⇐⇒ x ∈ (I − λV )−1 ⇐⇒ x = x1 (1, λ, λ2 , . . .) •

We need some auxiliary subspaces: 8. Definition If T ∈ BL(X, X) write 8.1

T −∞ (0) =

∞ [

T −n (0) and T ∞ (X) =

n=1

∞ \

T n (X)

n=1

for the hyperkernel and the hyperrange of T , and X \ 8.2 EX (T ) = (T − λI)−∞ (0) and FX (T ) = (T − λI)∞ (X) . λ∈C

λ∈C

Each of the subspaces in Definition 8 is linear, not necessarily closed, and hyperinvariant under T . We recall that T ∈ BL(X, X) is called algebraic if there is a nontrivial polynomial 0 6= p ∈ Poly for which 8.3

p(T ) = 0 ;

more generally T is said to be locally algebraic if [ 8.4 X = {p(T )−1 (0) : 0 6= p ∈ Poly} . For the record 4

9. Theorem If T ∈ BL(X, X) for a Banach space X then 9.1

T locally algebraic =⇒ T algebraic .

Necessary and sufficient for T to have finite descent is that 9.2

EX (T ) + T (X) = X .

Proof. The first part of this is known as Kaplansky’s Lemma; the proof [9] is a combination of Baire’s theorem and the Euclidean algorithm for polynomials. The Euclidean algorithm also gives equality [ 9.3 EX (T ) = {p(T )−1 (0) : 0 6= p ∈ Poly} = {x ∈ X : dim Poly(T )x < ∞} , and dually 9.4

FX (T ) =

\ {p(T )(X) : 0 6= p ∈ Poly} .

Then again with a combination of Baire’s theorem and the Euclidean algorithm, if T ∈ BL(X, X) there is ([12] Lemma 2.4) k ∈ N for which EX (T ) + T (X) = T −∞ (0) + T (X) = T −k (0) + T (X) •

9.5

Dually, using the Euclidean algorithm, we get half way: FX (T ) ∩ T −1 (0) = T ∞ (X) ∩ T −1 (0) .

9.6

For the essence of a possible spectral mapping theorem (cf [10]), we have 10. Theorem If S, T ∈ BL(X, X) satisfy ST = T S and either S ∈ {T k : k ∈ N}

10.1 or 10.2

V S − T U = I with {U, V } ⊆ comm(S, T ) ,

then there is equivalence 10.3

ST of finite ascent ⇐⇒ S , T of finite ascent ,

and equivalence 10.4

ST of finite descent ⇐⇒ S , T of finite descent .

Proof. The backward implications are easy ([7] Theorem 7.9.2): if S and T commute and satisfy S −k (0) = S −k−1 (0) and T −k (0) = T −k−1 (0) then (ST )−k (0) = S −k T −k (0) = S −k T −k−1 (0) = T −k−1 S −k (0) = T −k−1 S −k−1 (0) = (ST )−k−1 (0) , if instead ST = T S with S k X = S k+1 X and T k X = T k+1 X then (ST )k X = S k T k (X) = S k T k+1 X = T k+1 S k X = T k+1 S k+1 X = (ST )k+1 X . Also the forward implications are clear when (10.1) S = T k is a power of T ; if instead we assume (10.2) then we argue (ST )−1 (0) ⊆ T −1 (0) + T (X) and (ST )X ⊇ T −1 (0) ∩ T (X) ,

10.5

while if (U, V ) satisfies (10.2) then for arbitrary k ∈ N Vk S k − T k Uk = I with {Uk , Vk } ⊆ comm(S k , T k ) .

10.6 To verify (10.5) argue

ST x = 0 =⇒ x + T U x = V Sx ∈ T −1 (0) ; T (T x) = 0 =⇒ T x = T V Sx − T U T x = (ST )(V x) . For (10.6) note that for arbitrary k ∈ N V S − T U = I =⇒ V k+1 S k+1 − T U (I + V S + . . . + V k S k ) = I • 5

For an induced “spectrum” to be a closed set we have 11. Theorem T ∈ BL(X, X) is of finite descent then so is T − λI for sufficiently small λ ∈ C. Proof. This has been shown on Hilbert space by Han/Larson/Pan ([11] Lemma 2.2, Theorem 2.4). It is clear from the open mapping theorem (applied to the condition (4.4) with W = {0}) that if the condition (1.4) holds then also S −1 (0) + (T − U )(X) = Y whenever T − U ∈ BL(X, Y ) is sufficiently close to T ∈ BL(X, Y ): the problem is that we must also perturb S. However if S = T k and U = λI, so that EX (T − U ) = EX (T ), then we can argue EX (T − U ) + (T − U )(X) = EX (T ) + (T − U )(X) ⊇ S −1 (0) + (T − U )(X) = X • The subspaces of Definition 8 lead to certain special kinds of operator: 12. Definition We shall call T ∈ BL(X, X) triangular if the subspace EX (T ) is dense: 12.1

cl EX (T ) = X .

Dually T ∈ BL(X, X) is co-triangular if the subspace FX (T ) is trivial: 12.2

FX (T ) = {0} .

The shifts of Example 7 are either triangular or co-triangular: 13. Example On each of the spaces c0 and ℓp (1 ≤ p < ∞), the forward shift U is triangular, the backward shift V is co-triangular and the standard weight W is both triangular and co triangular. Proof. The hyperkernel of the backward shift is dense, since it includes all the “terminating” sequences: V −∞ (0) ⊇ c00 .

13.1 Thus

E(V ) ⊇ V −∞ (0) is dense

13.2 and also 13.3

F (V ) =

\

X

(V − λI)∞ (X) ⊇

(V − λI)−∞ (0) ⊇ V −∞ (0) is dense .

|λ|