GENERALIZED INVOLUTIONS ON BANACH SPACES

Manolis Magiropoulos

Abstract. We introduce generalized involutive operators on Banach spaces, based on the notion of a spectral family.

1. Introduction. In [7], among others, involutive operators of arbitrary order m, on a Banach space X, are examined (i.e. operators T: X → X, such that T^{m} = I for a positive integer m); it is proved that T = ∑_{j=1}^{m} z_{j}P_{j}, where {z_{j}} is the set of the m roots of 1, and P_{j} are projections on X such that P_{k}P_{j} = 0 for k ≠ j and ∑_{j=1}^{m} P_{j} = I. Thus, it is concluded that involutive operators are scalar on X (i.e. there exists a spectral measure μ(·) on X such that T = ∫_{ℝ} λdμ(λ)).

The above result can be derived immediately as an application of Proposition 10.6 in [4], can serve as a motivation to examine structural features of a slightly more general version of these operators, namely when {z_{j}} is an arbitrary finite subset of the unit circle. For evident reasons we call these (scalar) operators generalized involutive (g.i.) on X. There are two ways of spectrally classifying these operators; by use of a spectral measure or through the concept of a spectral family of projections. We shall follow the second approach. In Section 2 we characterize these operators as trigonometrically well–bounded deriving their spectral decomposition. (All pertinent theoretical tools are presented later on.) Similar results appear in Section 3 by constructing a normalized logarithm for g.i. operators, and by explicitly calculating also the corresponding “ergodic” operators following the formulas of the abstract setting of [2].

We present now, in brief, an account of the basic concepts and results needed throughout. By the term “operator on X” we will always refer to a bounded linear transformation with domain a (complex) Banach space X and range in X. We denote by B(X) the algebra of all operators on X and the prefix “s” denotes the strong topology of B(X). We take for granted the basic theory concerning spectral measures on Banach spaces. Supplementary theory concerning our machinery can also be found in [1,2,4].

\underline{Definition 1.1}. A spectral family in X is a uniformly bounded projection valued–function E(·): ℝ → B(X) which is s–right continuous on ℝ, has an s–left limit on ℝ and satisfies: (i) E(λ)E(μ) = E(μ)E(λ) = E(λ) for all λ, μ in ℝ with λ ≤ μ (ii) s–lim E(λ) = I (resp. 0) as λ → +∞ (resp. → −∞).

If there is a compact [a, b] ⊂ ℝ such that E(λ) = 0 for λ < a, E(λ) = I for λ ≥ b, we say that E(·) is concentrated on [a, b]. A theory of integration (in the Riemann–Stieltjes sense) with respect to spectral families is described in full detail in [4]. In particular for J = [a, b] and f in BV (J) we can define ∫_{J} fdE as an s–limit of sums, where the intermediate point in each partitioning subinterval is taken to be the right–end point of the subinterval. If, in addition, f is continuous on J, we can use arbitrary intermediate points [4].

The next definition is an equivalent characterization for a class of operators in B(X), that does not involve (directly) the original description (via the functional calculus introduced in [5] and [6]); it makes use of the integral previously defined. Let us introduce the symbol ∫_{J}^{⊕} fdE to describe the expression f(a)E(a)+∫_{[a,b]} fdE [1].

\underline{Definition 1.2}. Let T in B(X). T shall be called well–bounded of type (B), if there exists a compact interval J ⊂ R and a spectral family E(·) concentrated on J such that

T = ∫_{J}^{⊕} λdE(λ).

(In this case E(·) is unique and is called the spectral family of T.)

We next define a class of operators that lies in the core of our considerations. We use again an equivalent reformulation of the original definition, convenient for our purposes.

\underline{Definition 1.3}. An operator T in B(X) shall be called trigonometrically well– bounded if it can be expressed as

T = ∫_{[0,2π]}^{⊕} exp(iλ)dE(λ),

where J = [0, 2π] and E(·) a spectral family concentrated on J.

\underline{Remark}. We can always arrange, in Definition 1.3, E(2π^{−}) = I and then E(·) is uniquely determined and shall be simply called the spectral decomposition of T.

The next “multi–theorem” provides a convenient characterization for trigonometrically well–bounded operators, as well as the ground for constructing their “normalized” logarithm and the spectral families in both cases; it is a blending of results in [1] and [2] and it completes the main bulk of the theory needed throughout.

\underline{Theorem 1.4}. Let T be in B(X). Then

(i) T is trigonometrically well–bounded if and only if T = exp(iS), for a well– bounded operator S of type (B) on X.

(ii) If T is trigonometrically well–bounded, then there is a (unique) well–bounded operator A of type (B) on X, such that T = e^{iA}, σ(A) ⊂ [0, 2π] and 2π is not in the point spectrum of A. (Notation A = Arg T, argument of T.)

(iii) If T and A are as in (ii), then the spectral decomposition of T coincides with

the spectral family of A. (iv) If T is as in (ii) and it is also “power–bounded” on a reflexive Banach space

X, i.e. sup{∥T^{n}∥ : n ∈ ℤ} < ∞, then the argument of T is given by

Arg T = πI − πQ_{0} + iB_{0},

where

Q_{0} = s − lim_{n} \frac{1}{n} ∑_{k=0}^{n−1} T^{k} and B_{0} = s − lim_{n} ∑_{k=−n}^{n}′ \frac{T^{k}}{k}

exist in B(X) (and the “superscript prime” for the second series indicates omission of n = 0).

(v) If T is trigonometrically well–bounded and power–bounded on a reflexive Banach space X, then the spectral decomposition of T is given by

E(λ) = \frac{1}{2πi}\{iλI − B_{λ} + B_{0}\} + \frac{1}{2}\{Q_{λ} + Q_{0}\} for 0 ≤ λ < 2π,

where

Q_{λ} = s − lim_{n} \frac{1}{n} ∑_{k=0}^{n−1} exp(−ikλ)T^{k}

and

B_{λ} = s − lim_{n} ∑_{k=−n}^{n}′ exp(−ikλ)\frac{T^{k}}{k}

are (automatically) in B(X).

\underline{Remarks}. 1) In Theorem 1.4(v) the reflexivity of X is used in order to ensure that Q_{λ} is in B(X) (by use of the theory of V.III.5 in [3]); since for g.i. operators we can easily check – which actually we will see later – that this is true on any Banach space, we have not required reflexivity (see Lemma 3.1). 2) In case X is a Hilbert space, we see that g.i. operators on X are simply unitary on X (the corresponding projections taken as self–adjoint). This is not accidental. In fact, any power–bounded operator on a Hilbert space is similar to a unitary operator on the same space, and certainly our operators are power–bounded since

T^{k} = ∑_{j=1}^{m} z_{j}^{k}P_{j}

and thus,

sup{∥T^{k}∥ : k ∈ ℤ} ≤ ∑_{j=1}^{m} ∥P_{j}∥ < ∞.

(Scholium: The above mentioned remarkable feature of power bounded operators on a Hilbert space – which is a consequence of deep theorems due to Sz–Nagy has also a rather elementary proof (e.g. see J.A. Van Casteren’s “Generators of Strongly Continuous Semigroups,” Research Notes in Mathematics, No. 115, Pitman, 1985)).

2. G.I. Operators as Trigonometrically Well–Bounded Operators. Throughout this section let

T = ∑_{j=1}^{m} z_{j}P_{j},

for an arbitrary but fixed positive integer m; to avoid unnecessary fiddling, we shall assume T ≠ I (i.e. we shall avoid the case m = 1, z_{1} = 1).

Note that if T = I its spectral decomposition is 0 for λ < 0 and I for λ ≥ 0.

Now let θ_{j} be the principal argument of z_{j} in [0, 2π). Based on the theory of integration with respect to a spectral family is not hard to guess that the natural candidate for the spectral decomposition of T is given by

E(λ) = ∑_{θ_{j}≤λ}^{m} P_{j}, for each λ ∈ R,  (∗)

where the lower index symbolism indicates summation with respect to all j, 1 ≤ j ≤ m, such that θ_{j} ≤ λ (with the convention ∑_{∅} = 0).

\underline{Remark}. By definition, it is evident that E(λ) = 0 for λ < 0 and E(λ) = I for λ ≥ 2π.

The two lemmas which follow will demonstrate that the above claim for the spectral decomposition of T is true. We shall denote convergence with respect to the uniform topology of B(X) by “u”.

\underline{Lemma 2.1}. The function E(·) defined on ℝ via (∗) is a spectral family (concentrated on [0, 2π]).

\underline{Proof}. Evidently, each E(λ) is a projection on X, and since

sup_{λ} ∥E(λ)∥ ≤ ∑_{j=1}^{m} ∥P_{j}∥ < ∞,

they form a uniformly bounded family on ℝ. Property (i) of Definition 1.2 is immediate by the properties of the P_{j}’s and (ii) is a direct consequence of the preceding remark (and it actually holds in the u–sense). We now need only to demonstrate s–right continuity and existence of the s–left hand limit of E(λ) at each λ in ℝ. For a simpler notation we do that by showing, respectively,

(a)  u − lim_{ϵ→0^{+}} E(λ + ϵ) = E(λ)

and

(b)  u − lim_{ϵ→0^{+}} E(λ − ϵ) = ∑_{θ_{j}<λ} P_{j}.

For (a) it is enough to observe that if ϵ > 0, then

E(λ + ϵ) − E(λ) = ∑_{λ<θ_{j}≤λ+ϵ} P_{j} \overrightarrow{ϵ→0^{+}} ∑_{∅} = 0.

Similarly working for (b), we have

E(λ − ϵ) = E(λ) − ∑_{λ−ϵ<θ_{j}≤λ} P_{j} \overrightarrow{ϵ→0^{+}} E(λ) − ∑_{θ_{j}=λ} P_{j} = ∑_{θ_{j}<λ} P_{j}.

\underline{Lemma 2.2}. For T as before and E(·) defined via (∗), we have
T = ∫_{[0,2π]}^{⊕} exp(iλ)dE(λ)

(as a strong, and in fact, uniform limit of Riemann–Stieltjes sums). \underline{Proof}. It is enough to show that

T = u − lim_{ϕ} {E(0) + ∑_{k=1}^{n} exp(iλ_{k})(E(λ_{k}) − E(λ_{k−1}))},

over all partitions ϕ = {λ_{0}, …, λ_{n}} of [0, 2π].

Note at first that E(0) corresponds to 0 if z_{j} ≠ 1 for all j, and to P_{j_{0}} if z_{j_{0}} = 1 for some (unique) 1 ≤ j_{0} ≤ m. Observe next that

∑_{k=1}^{n} exp(iλ_{k})(E(λ_{k}) − E(λ_{k−1})) = ∑_{k=1}^{n} exp(iλ_{k})( ∑_{λ_{k−1}<θ_{j}≤λ_{k}} P_{j}).

Since

max_{1≤k≤n} |λ_{k} − λ_{k−1}|

becomes very small as n becomes large, then

∑_{λ_{k−1}<θ_{j}≤λ_{k}} P_{j}

is either 0 or P_{j_{0}} (whenever θ_{j_{0}} = λ_{k}), and the result follows.

We summarize the results of the two lemmas in the following theorem.

\underline{Theorem 2.3}. Let

T = ∑_{j=1}^{m} exp(iθ_{j})P_{j},

for a (fixed) m ≥ 1, where θ_{j} are distinct in [0, 2π) and P_{j} are projections on a (fixed) Banach space X, such that P_{k}P_{j} = 0 for k ≠ j and

∑_{j=1}^{m} P_{j} = I (the identity on X).

Then (i) T is trigonometrically well–bounded on X, and its spectral decomposition is given by

E(λ) = ∑_{θ_{j}≤λ} P_{j}, for each λ in ℝ.

(ii) T possesses as argument the operator

A = ∑_{j=1}^{m} θ_{j}P_{j} whose spectral family is E(·).

\underline{Proof}. (i) We need only to check that E(2π^{−}) = I, and appeal to the remark following Definition 1.3.

(ii)  ∫_{[0,2π]}^{⊕} λdE(λ) = Arg T

by Definition 1.2, Definition 1.3, Theorem 1.4 (ii)–(iii), and the result in (i). Repeating verbatim the proof of Lemma 2.2, we see that

u − ∫_{[0,2π]}^{⊕} λdE(λ) = ∑_{j=1}^{m} θ_{j}P_{j}.

\underline{Remark}. It is natural at this point to set the question of constructing the spectral decomposition of T (or of the argument of T), without departing from an intuitive observation. First note that for detecting Arg T, an alternative approach could have been the following. Let

A = ∑_{j=1}^{m} θ_{j}P_{j}.

We easily check that T = exp(iA) and σ(A) ⊂ [0, 2π], since σ(A) = {θ_{j} : 1 ≤ j ≤ m}. To check that A is well–bounded of type (B) without of course involving the previous construction of E(λ), we can appeal to the equivalent definition in [1] involving a (weakly compact) AC([0, 2π]) functional calculus.

The next section reveals how, by rather elementary calculations that involve classical Fourier series we can reproduce, using the formula of Theorem 1.4 (iv)–(v), the results of Section 2.

3. The Construction of the Argument and the Spectral Decomposition of G.I. Operators. Set

A = ∑_{j=1}^{m} θ_{j}P_{j}.

Since T = exp(iA) and A is well–bounded of type (B), we conclude that T is trigonometrically well–bounded without appealing to the results of the lemmas in Section 2. Since T is also power–bounded, Theorem 1.4 (iv)–(v) is applicable. We (temporarily) fix λ in ℝ and proceed to calculate the operators Q_{λ}, B_{λ}, in the subsequent lemma. For notational convenience, we set P_{j,λ} to be 0 if θ_{j} ≠ λ and P_{j_{0}} if θ_{j_{0}} = λ, for a (unique) j_{0}, (1 ≤ j_{0} ≤ m).

We also denote by E(λ) the operator

∑_{θ_{j}≤λ} P_{j}.

\underline{Lemma 3.1}. Let T be as in Section 2. Then the operators Q_{λ} and B_{λ} defined in Theorem 1.4 (v) have the following representation, for any λ in ℝ.

(i) Q_{λ} = P_{j,λ} (ii) B_{λ} = i{(λ + π)I + πP_{j,λ} − 2πE(λ) − A}.

\underline{Proof}. (i) It is immediate that

Q_{λ} = ∑_{j=1}^{m} ω_{j}P_{j},

where

ω_{j} = lim_{n} ω_{nj}, ω_{nj} = \frac{1}{n} ∑_{k=0}^{n−1} exp(ik(θ_{j} − λ)).

Evidently,

ω_{nj} = \{ \begin{matrix}1, & if θ_{j} = λ \\ \frac{1}{n}\{1 − exp(i(θ_{j} − λ))\}, & if θ_{j} ≠ λ, \end{matrix}

and thus,

ω_{nj} = \{ \begin{matrix}1, & if θ_{j} = λ \\ 0, & otherwise. \end{matrix}

(ii) Similarly, due to the symmetric summation, we can easily see that

B_{λ} = ∑_{j=1}^{m} δ_{j}P_{j},

where

δ_{j} = s − lim_{n} 2i ∑_{k=1}^{n} \frac{sin k(θ_{j} − λ)}{k} = 2is − lim_{n} ∑_{k=1}^{n} \frac{sin k(θ_{j} − λ)}{k}.

Based on the classical formula for the Fourier series of

\frac{π − θ}{2}

for 0 < θ < 2π, we can easily see that

δ_{j} = \{ \begin{matrix}0, & for 0 ≤ λ = θ_{j} < 2π \\ i(λ − θ_{j} + π), & for λ < θ_{j} < 2π \\ i(λ − θ_{j} − π), & for 0 ≤ θ_{j} < λ. \end{matrix}

We conclude that B_{λ} = λ(I − P_{j,λ} − (A − λP_{j,λ}) − π(E(λ) − P_{j,λ}) + π(I − E(λ)), and the result follows.

\underline{Theorem 3.2}. Let T be an operator as in Theorem 2.3. Then this trigonometrically well–bounded operator has an argument, Arg T, and a spectral decomposition, F (·), such that

(i) Arg T = A (ii) F (λ) = E(λ) for each λ in ℝ.

\underline{Proof}.

(i) Theorem 1.4 (iv) implies Arg T = πI − πP_{j,0} − (πI − πP_{j,0} − A) = A, since E(0) = Q_{0} = P_{j,0}.

(ii) F (λ) = E(λ) for λ < 0 or λ ≥ 2π is immediate, since F (·) is concentrated on [0, 2π] and E(·) behaves, by construction, in a similar way. Now let 0 ≤ λ < 2π. Then by Theorem 1.4 (v),

F (λ) = \frac{1}{2πi}\{iλπI − B_{λ} + B_{0}\} + \frac{1}{2}\{Q_{λ} + Q_{0}\}.

Substituting B_{λ} and B_{0} as given in Lemma 3.1 (ii) we can easily obtain the announced equality.

\underline{References}

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Manolis Magiropoulos General Department of School of Technological Applications Technological Educational Institute of Heraklion Stavromenos, Heraklion 71500 Crete, Greece email: mageir@stef.teiher.gr