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Approximation by Unitary and Essentially Unitary Operators

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Approximation by Unitary and Essentially Unitary Operators Acta Sci. Math., 39 (1977), 141—151 Approximation by unitary and essentially unitary operators DONALD D. ROGERS In troduction. In [9] P. R. HALMOS formulated the problem of normal spectral approximation in the algebra of bounded linear operators on a Hilbert space. One special case of this problem is the problem of unitary approximation; this case has been studied in [3], [7, Problem 119], and [13]. The main purpose of this paper is to continue this study of unitary approximation and some related problems. In Section 1 we determine the distance (in the operator norm) from an arbitrary operator on a separable infinite-dimensional Hilbert space to the set of unitary operators in terms of familiar operator parameters. We also study the problem of the existence of unitary approximants. Several conditions are given that are sufficient for the existence of a unitary approximant, and it is shown that some operators fail to have a unitary approximant. This existence problem is solved completely for weighted shifts and compact operators. Section 2 studies the problem of approximation by two sets of essentially unitary operators. It is shown that both the set of compact perturbations of unitary operators and the set of essentially unitary operators are proximinal; this latter fact is shown to be equivalent to the proximinality of the unitary elements in the Calkin algebra. Notation. Throughout this paper H will denote a fixed separable infinite- dimensional complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an arbitrary operator T, we write j|T"[| = sup {||Tf\\ :f in //and ||/|| = 1} and m{T)=mi {|| 7/11:/ in H and ||/|| = 1}. The spectral radius of T is r{T). We write \T\=(T*Tf>2, and E(-) is the spectral measure for |r|. The index of an operator T is defined by ind (T)=dim ker (T)—dim ker (T*) if at least one of these numbers is finite, and we use the convention that ind (T)—0 if both number are . Received October 30, 1975, in revised form June 9, 1976. This paper is part of the author's Ph. D. thesis written at Indiana University under the direc- tion of Professor P. R. Halmos. 142 Donald D. Rogers The ideal of compact operators is denoted by K(H), and n is the canonical homomorphism from B(H) onto the Calkin algebra C(H)=B(H)/K(H). The operator T is Fredholm if n(T) is invertible in C(H). The spectrum of n(T) is oe(T) with spectral radius re(T)\ the complement of oe(T) is denoted by , Qe(T). We write ||r[|e=[|Tr(7 )|| and me(J)= theinfimum of ae{\T\). The unilate- ral weighted shift of multiplicity one with weight sequence (a^aa, ...) is denoted shift (^,«2, ...). If Jl is a set of operators, then an operator X0 in M is an Jt-approximant of the operator T if Hr-A^l =inf (HT-Al :X in J(). The set Jl is proximinal in B(H) (or simply proximinal) if every operator T has an J(- approximant. 1. Unitary operators. We shall frequently use the following theorem. It appears in [5, Theorem 2.2] for the case that 7\ is a Fredholm operator; a slightly different proof is given below for completeness. 1.1. Theorem. 7/ Tx and T2 are in B(H) and if W^-T^^m^T^, then ind (J^ind (J;). Proof. Consider first the case Tx = 1. If ||1 —r2||e<l, then there exists K in K(H) such that ||1 -T2-K\\<\. Hence T2+K is invertible, so clearly ind(7'2 + A') = =0 Thus T2 is Fredholm and ind(r2) = ind (T2 + K) = 0 [2, Lemma 5.20]. Thus ind (r2)=ind(l)=0. For the general case, we can assume that we(7i)>0. Then there exists L in B(H) such that ||Z,||c=l/we(r1) and LTt is a compact perturbation of the identity (this can be seen by looking at the polar decomposition of 7\). Then ¡| 1 —LT2\\e= T , , = ||Lr1-L7 2||e^||Z||e-||7 1-7 2[|e<l. Hence LT2 is Fredholm of index 0 by the above result. Consequently Tt is Fredholm if and only if T2 is Fredholm, and in this case ind (7i)= — ind (L)=ind (T2) by the additivity of the index for Fredholm operators [2, Theorem 5.36]. If both Tx and T2 fail to be Fredholm, then dim ker T* = X0=dim ker T2 . This follows because both LTX and LT2 are Fredholm, which implies that both 7\ and T2 have closed range and finite-dimensional kernel [2, proof of Theorem 5.17]. Hence both 7\ and T2 are Fredholm unless dim ker 7\* = K0=dim ker T2 . Thus 1 in this one remaining case it follows that ind (7 1)=ind (T2)= — 1.2. Corollary. If ind (r)<0 and U is a unitary operator, then \\T— £/||s Proof. Clearly ||r-C/|| = ||i7*r-l||Sr(C/+r-l). Assertion: Each number in the open ball {C:!C[-=wc(7^)} is an eigenvalue of T* U. To see this, let %\<me(T) and apply Theorem 1.1 to the operators T1 = U*T and T2 = U' T— £; notice Approximation by unitary and essentially unitai y operators 145 that me(T)=me{U* T) and ind (U* T)=ind(T). Hence ind (U* T-Q=ind(U* T)<0. This proves the assertion, and the assertion implies 1.2. We can now determine the distance from an arbitrary operator T to the set of unitary operators. Write u{T) — 'mi{\T—U\\:U a unitary operator}. 1.3. Theorem. (i)// ind(r)=0, then M(r)=max{||T||-l,\-m(T)}. (ii) If ind(r)<0, then w(r)=max {||71 -1, 1 +m,(T)}. (The case ind(T)>0 follows from (ii) by considering the adjoint of T). Proof. Assertion (i) is true also in finite dimensions [3] and is proved here in a similar manner. The main point is that it is possible to find a unitary operator U such that T=U\T\ by enlarging the partial isometry in the polar decomposition of T (if necessary). Then ||T— C/|| = |]|T|-1||, and it is easy to see that |[ |r|-11| = =max {j|71 — 1, 1 —m(T)}. That this maximum is a lower bound for u(T) is also easy to see by using the triangle inequality. This proves assertion (i). To prove assertion (ii), let E(-) be the spectral measure for \T\, and for e>0 let Ec denote the projection £([0, me(T) + c]). Then dim EC(H) = X0 since me(T) has the equivalent definition me(7')=inf {x^0:dim i?([0, x])7/=X0} (see [4, p. 185]). Because ind(J')<0, there exists a (non-unitary) isometry S such that T—S\T\. Because EE(H) and ker S*© SEE(H) have equal dimension and co-dimension, there exists an isometry VE in 5(H) that maps EC(H) onto ker S*®SEE(H). Define the operator UT-= VTE, +S{\-EE). A s s e r t i o n: UE is a unitary operator. Proof. It is easy to see that UE is an isometry; that UE is onto follows since UE(EE(H)) = ker S*®SEC(H) and UT{HQET{H)) = S(HQ E,(H)). Assertion: |]T— C/e||^max{||Tj| — 1, 1 +me(T)+ 6}. Proof. Clearly ||r-«7J=||[/£*r-l||; we examine the operator U*T. It is not difficult to see from the definition of UE that EC(H) reduces U*T=U*S\T\. With respect to the decomposition H=ET(H)@(L-ET) (H), if follows that U*T= =XC®YC with \\XE\\ SME(T)+E and YC=restriction of |r| to the (reducing) subspace (1 — EC) (H). Thus ||t/*r-l||=max{i|Y£-l||,||y£-l||}. Clearly ||jT,-l|l Si +me(T)+ £ and ||r,-l||s|||r|-l||. The fact that max {l+me(T)+€, |||J|-l||}=max {1 + +me(T)+£, liril-1} follows easily. This proves u(T)zimax {1 +me(T), ||r||-l}. 144 Donald D. Rogers The reverse inequality follows from Corollary 1.2 and the triangle inequality. This proves Theorem 1.3. In [8] it was shown that every operator has a positive approximant that is in the C*-algebra generated by the identity and the operator. For approximation by unitary operators, however, the situation is considerably different. 1.4. Theorem. (i) If ind (r)=0, then T=U\T\ for some unitary approximant U. If the index of T is non-zero, then u(T)s 1 ; we consider the following two cases. (ii) If ind (T)^0 and u(T) = 1, then T fails to have a unitary approximant. (in) If ind(r)<0 and w(r)>l, then each one of the following conditions is suffcient for T to have a unitary approximant: (a) ||7||-l>l+me(r), (b) dim E ([0, me(7)])(77) = K0, (c) me(T) is a cluster point of eigenvalues of \ T\. (The case ind (7")=>0 and u(T)>\ follows from (iii) by considering the adjoint of T). Proof. Assertion (i) follows easily from the proof of Theorem 1.3 (i). Assertion (ii) is a consequence of [14, p. 408]. For if ind^^O and U is a unitary operator such that \\U—T\\=u(T) = \, then || 1-17*711 = 1 and hence [14] implies ind ((1 -£/*r)-l)=0=ind (-U*T).
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