Elementary properties of 1-isometries on a Hilbert space

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1 Eleentary properties of 1-isoetries on a Hilbert space Muneo Chō Departaent of Matheatics, Kanagawa University Hiratsua , Japan Caixing Gu y Departent of Matheatics, California Polytechnic State University San Luis Obispo, CA 93407, U.S.A. Woo Young Lee z Departent of Matheatics, Seoul National University Seoul , Korea Deceber 8, 2015 Abstract Inspired by recent wors on -isoetries for a positive integer ; in this paper we introduce the classes of 1-isoetries and 1-unitaries on a Hilbert space. We show that an 1-isoetry on a nite diensional coplex Hilbert space H with diension N is in fact an (2N 1)-isoetry. We describe the spectra of such operators, study the quasinilpotent perturbations of 1-isoetries and characterize when tensor products of 1-isoetries are also 1-isoetries. As a surpring byproduct, we obtain a generalization of Nagy-Foias-Langer decoposition of a contraction into an unitary and a copletely nonunitary contraction. 1 Introduction Since a systeatic study on -isoetries by Agler and Stanus [3], [4] and [5], the theory of -isoetries has been highly developed. The theory for -isoetries on Hilbert spaces has rich connections to Toeplitz operators, classical function theory, and other areas of chiyo01@anagawa-u.ac.jp y cgu@calpoly.edu z wylee@snu.ac.r 1

2 atheatics. In particular, a class of 2-isoetries arises fro non-stationary stochastic process of Brownian otion (see [4]). The wor of Richter [27] and [28] on analytic 2- isoetries has a connection with the invariant subspaces of the shift operator on the Dirichlet space, see also related papers [24], [25] and [30] in this direction. On the other hand, the de nition of -isoetries depends on the degree of the polynoial (yx 1) in two variables. Thus we ay as what happens if! 1. This stiulates a new notion of 1-isoetries. The ai of this paper is to explore eleentary properties of 1-isoetries. In particular, we give a coplete characterization of 1-isoetries on a nite diensional Hilbert space. Let H be a coplex Hilbert space and let B(H) denote the set of all bounded linear operators acting on H: If T 2 B(H); we write (T ) and ap (T ) for the spectru and the approxiate spectru of T; respectively. If H 0 is a subspace of H; P H0 is the projection fro H onto H 0 : An operator T 2 B(H) is called an -isoetry for a positive integer (as in Agler and Stanus [3]) if Note that (T ) : (yx 1) j yt ; xt ( 1) 0 ( 1) 0 T T 0: y x j yt ; xt +1 (T ) T (T )T (T ): (1) Thus if T is an -isoetry, then T is an n-isoetry for all n : We say T is a strict -isoetry if T is an -isoetry but not an ( 1)-isoetry. We now introduce the class of 1-isoetries. De nition 1.1 Let T 2 B(H): The operator T is called an 1-isoetry if li sup (T ) 1 0:!1 Also T is called a nite-isoetry if T is an -isoetry for soe 1: If H 0 is an invariant subspace of T, then (T jh 0 ) P H0 (T )jh 0 (2) where T jh 0 is the restriction of T to H 0 : This property is one of the ain otivations of hereditary functional calculus as in [1], [2], [3] and [20]. So, if T on H is an 1-isoetry, then T jh 0 is also an 1-isoetry. The rst otivation of studying 1-isoetries coes fro recent interests in -isoetries on Hilbert spaces, Banach spaces and etric spaces [6], [8], [14], [18], [21], [26], and [31]. The second otivation is that 1-isoetries see to be natural liits of -isoetries as! 1; see Proposition 4.6 below. However, the ain otivation is that 1-isoetries 2

3 enjoy any properties of -isoetries, as we will deonstrate in this paper. This new class of operators also poses soe interesting questions and challenges. For exaple, one of the ain tools in [3], [6] and [28] for studying an -isoetry T is the nonnegative covariance operator 1 (T ) which is not available in an 1-isoetry. We hope that this study will deepen our understanding of -isoetries. First we give soe exaples. Recall a unilateral weighted shift T on l 2 is de ned by T e j w j e j+1 ; j 0, where fe j ; j 0g is the standard basis of l 2. Exaple 1.2 Let Q be the weighted shift on l 2 with weights w j 1 : Then Q is quasinilpotent and T I + Q is an 1-isoetry by Theore 4.4 below. But T is not a nite-isoetry. j+1 A direct coputation of showing (T ) 6 0 for any 1 sees to be di cult. Note that (T ) f1g : Recall that if S is an -isoetry and (S) consists of a nite nuber of points, then S is the direct su of operators of the for I + Q 0 ; where Q 0 is a nilpotent operator, see Proposition 11 in [17]. Hence if T I + Q is a nite-isoetry, then Q is a nilpotent operator, which is a contradiction. Exaple 1.3 Let T n be an n n Jordan bloc 2 T n 6 4 n 1 0 n n 0 0 n ni n + 1 n J n; where j n j 1 and J n : Then, by a direct calculation (for exaple, using the rst forula in Lea 4.1 below with T n I n and Q 1 n J n), 2n 1 (T n ) 0 and 2n 2 (T n ) 2n 2 n 1 1 n 2 n 1 (n 1) n J n T (n 1) (n 1) J n 6 0: That is, T n is a strict (2n 1)-isoetry. Let T T 1 T 2 T 3 : Then T is an 1-isoetry but not a nite-isoetry. Furtherore, (T ) f n : n 1g. To see this, let S n T 1 T n n+1 I n+2 I : 3

4 Then S n is a strict (2n 1)-isoetry and S n S l S l S n for all n; l 1: It is also clear that S n! T in operator nor. Therefore, by Proposition 4.6 below, T is an 1-isoetry. Clearly, T is not a nite-isoetry since for any 1: (T ) (T 1 ) (T 2 ) (T 3 ) The 1-isoetric atrices In this section, we discuss basic spectral properties of an 1-isoetry. As applications of those properties, we show that if H is a nite diensional Hilbert space, then T 2 B(H) is an 1-isoetry if and only if T is an -isoetry for soe 1: The spectru of an -isoetry was described in [3] on a Hilbert space and in [6] on a Banach space. Siilarly, we have the following generalization. Let D be the open unit dis be the unit circle. Proposition 2.1 If T is an 1-isoetry, then ap (T Therefore, either (T ) D or (T In particular, T is left invertible. Proof. Let 2 ap (T ) and let h i 2 H be a sequence of unit vectors such that (T I) h i! 0 as i! 1: It is clear that T h i 2! jj 2 as i! 1: Hence T h (T )h i ; h i i ( 1) h i 2 0! ( 1) jj 2 (jj 2 1) : 0 Thus (T ) jh (T )h i ; h i ij and hence; (T ) 1 jj 2 1 : Therefore li sup!1 (T ) 1 0 iplies that jj 1: By Exaple 1.3, any copact set could be the spectru of an 1-isoetry. De nition 2.2 We say T is an 1-unitary if both T and T are 1-isoetries. Siilarly, for 1; T is an -unitary if both T and T are -isoetries. T is a nite-unitary if T is an -unitary for soe 1: Corollary 2.3 If T is an 1-unitary, then (T Proof. The proof is the sae as the proof of Corollary 1.22 in [3]. We prove by contradiction. If (T ) then by Proposition 2.1, (T ) D and 0 2 ap (T ): Thus T is onto and 4

5 0 2 ap (T ): By applying Proposition 2.1 to T ; we have (T This is a contraction to (T ) D : Note that exaples in Exaple 1.2 and Exaple 1.3 are 1-unitaries. We next show that the following result about eigenvalues for an -isoetry does not extend to 1-isoetries, see part (a) of Theore 1 in [15]. The operator in this exaple is just the adjoint of the operator in Exaple 1.2. Theore 2.4 [15] Let T be an -isoetry. If is an eigenvalue of T; then is an eigenvalue of T : Siilarly, if 2 ap (T ); then 2 ap (T ): Exaple 2.5 Let T I + Q ; where jj 1 and Q is the weighted shift on l 2 with weights w j 1 : Then T is an 1-isoetry. Furtherore, (T I) e j+1 0 Q e 0 0; but T I e 0 e 1 6 0: It is easy to see that is an eigenvalue of T; but is not an eigenvalue of T : However a siilar result for eigenvectors or approxiate eigenvectors of an -isoetry in Theore 1 of [15] does extend to an 1-isoetry. See also Lea 19 in [2] for a related result. Proposition 2.6 Let T be an 1-isoetry. (a) Eigenvectors of T corresponding to distinct eigenvalues are orthogonal. (b) If and are two distinct approxiate eigenvalues of T; and fx n g and fy n g are sequence of unit vectors such that (T I)x n! 0 and (T I)y n! 0; then hx n ; y n i! 0: Proof. We rst prove (a). Let and be two distinct eigenvalues of T. By Proposition 2.1, jj jj 1; so 1 6 0: Let x and y be two unit vectors such that (T I)x 0 and (T I)y 0: Then h (T )x; yi T ( 1) x; T y ( 1) hx; yi ( 1) hx; yi : 0 0 It follows that j( 1) hx; yij 1 jh (T )x; yij 1 (T ) 1 : By taing lisup as! 1 of the inequality above, we see that hx; yi 0: The proof of (b) is siilar. Let and be two distinct approxiate eigenvalues of T. By Proposition 2.1, jj jj 1; so 1 6 0: Let fx n g and fy n g be two sequences of unit vectors such that (T I)x n! 0 and (T I)y n! 0: To prove hx n ; y n i! 0; let 5

6 xnj ; y nj be any convergent subsequence of hxn ; y n i such that x nj ; y nj! a; we shall show that a 0: Note that for each x 1; j( 1) aj li ( 1) x nj ; y nj n j!1 ( 1) li xnj ; y nj n j!1 0 ( 1) li T x nj ; T y nj n j!1 0 li (T )x nj ; y nj (T ) : Therefore n j!1 j 1j li!1 jaj1 li sup j( 1) aj 1!1 li sup (T ) 1 0:!1 This iplies a 0 and hence hx n ; y n i! 0: We next show that the above proposition can be signi cantly strengthened. We rst need a lea. Lea 2.7 For any two coplex nubers and, the following holds: (T ) (T I) 1 T 1 2 (T I) 2 ( 1) 3 : 1 ; 2 ; Proof. Using the ultinoial forula, (T ) (yx 1) j yt ; xt jyt (y ) x + (x ) + ( 1) ; xt (y ) 1 x 1 2 (x ) 2 ( 1) 3 j yt ; 2 ; ; xt 3 (T I) 1 T 1 2 (T I) 2 ( 1) 3 : 1 ; 2 ; The proof is coplete. Lea 2.8 Let T be an 1-isoetry. If and are two distinct eigenvalues of T; then er(t I)? er(t I) l for all ; l 1: 6

7 Proof. We prove the lea by induction. For the case l 1. Let v 1 ; v 2 be unit vectors such that (T I)v 1 (T I)v 2 0. Then by Lea 2.7, h (T )v 1 ; v 2 i * ( 1) hv 1 ; v 2 i ; 1 ; 2 ; 3 ( 1) 3 T 1 2 (T I) 2 v 1 ; (T I) 1 v 2 + since (T I) 2 v 1 (T I) 1 v 2 0 for all 1 ; By assuption 6 1, thus li sup j(!1 1) hv 1 ; v 2 ij 1 li sup jh (T )v 1 ; v 2 ij 1!1 li sup (T ) 1 0; (3)!1 which gives hv 1 ; v 2 i 0. We now x l 1 and use induction on : Assue er(t I)? er(t I): Let v 1 2 er(t I) +1, v 2 2 er(t I). We will show that hv 1 ; v 2 i 0, and hence er(t I)? er(t I) for all 1. Note that h (T )v 1 ; v 2 i * * ( 1) 3 T 1 2 (T I) 2 v 1 ; (T I) 1 v 2 1 ; 2 ; 3 + ( 1) 3 2 (T I) 2 v 1 ; v 2 0; 2 ; 3 ( 1) hv 1 ; v 2 i ; since if 1 1, (T I) 1 v 2 0. Moreover, if 2 + 1, (T I) 2 v 1 0; and if 1 2, (T I) (T I) 2 v 1 (T I) + 2 v 1 0, so (T I) 2 v 1 2 er(t I), thus (T I) 2 v 1? v 2 and h(t I) 2 v 1 ; v 2 i 0 by the inductive hypothesis. Thus the only ter left is when 1 2 0; 3. An arguent siilar to (3) shows that hv 1 ; v 2 i 0: This proves that By syetry, we also have er(t I)? er(t I) for all 1: (4) er(t I)? er(t I) l for all l 1: (5) We will show that er(t I)? er(s I) l for any ; l 1 by using induction on l. Assue for xed l; er(t I)? er(t I) l for all 1: (6) 7

8 We now show that er(t I)? er(t I) l+1 for all 1 as well. We do this by using induction on : For 1; this is just (5). Now we assue We will show er(t I) n? er(s I) l+1 : (7) er(t I) n+1? er(t I) l+1 : Let v 1 2 er(t I) n+1 and v 2 2 er(t I) l+1. A siilar coputation yields: 0 h (T )v 1 ; v 2 i * + ( 1) 3 T 1 2 (T I) 2 v 1 ; (T I) 1 v 2 : (8) 1 ; 2 ; Now, if 1 1 and 2 0, (T I) 1 v 2 2 er(t I) l and (T I) 2 v 1 2 er(t I) n. Thus by (6), these ters in (8) are zero. If 1 0 and 2 > 0, (T I) 1 v 2 2 er(t I) l+1 and (T I) 2 v 1 2 er(t I) n. Then by the inductive hypothesis (7), these ters (8) are also zero. The only ter left in (8) is when 1 2 0; 3. Finally, an arguent siilar to (3) shows that hv 1 ; v 2 i 0: Rear 2.9 Siilarly, if T is an 1-isoetry, and if and are two distinct approxiate eigenvalues of T; and fx n g and fy n g are sequence of unit vectors such that (T I) x n! 0 and (T I) l y n! 0 for soe xed ; l 1; then hx n ; y n i! 0: Theore 2.10 Assue H is a nite diensional coplex Hilbert space with di(h) N: Then T 2 B(H) is an 1-isoetry if and only if T is an (2N 1)-isoetry. Proof. Assue T is an 1-isoetry on a nite diensional H. By Proposition 2.1, it is (T Let p() Q i1 ( i) p i be the characteristic polynoial of T; where i are distinct eigenvalues of T: By Lea 2.8, T is unitarily equivalent to direct su T M ( i I + Q i ) i1 where Q i is a nilpotent atrix of order p i : A direct calculation (or by Lea 4.1) shows that (T ) 0 for ax f2p i 1; i 1; 2; ; g : An -isoetry on a nite diensional Hilbert space H is the direct su of operators of the for I + Q as in [2] where jj 1 and Q is a nilpotent atrix. See [2] also for ore general hereditary roots than just -isoetries on a nite diensional H: 8

9 3 The decoposition of an 1-isoetry In this section, for an 1-isoetry T, we identify invariant subspaces of T such that the restriction of T to those invariant subspaces are nite-isoetries. In particular, we decopose an 1-unitary T into direct su of -unitaries for 2 f1; 2; 3; g [ f1g. The rst result identi es soe invariant subspace H for an invertible T such that T jh is an -isoetry. Let H : \ er( n (T )): It follows fro the de nition that n H 1 H 2 H 3 : Proposition 3.1 Let T 2 B(H). Assue er(t ) f0g : Then H is an invariant subspace for T and T jh is an -isoetry. Proof. We rst prove H is invariant for T: If x 2 H ; then for any n ; Then by (1), n (T )x 0 and n+1 (T )x 0: [ n+1 (T ) + n (T )] x T n (T )T x 0: Since T is injective, n (T )T x 0 for any n : That is, T x 2 H : Now by (2), Therefore, T jh is an -isoetry. (T jh ) P H (T )jh P H 0 0: If T is an -isoetry, then n (T ) 0 and er( n (T )) H for n ; so we have the following result. Corollary 3.2 If T 2 B(H) is an invertible -isoetry, then for n 1; 2; : : : ; 1, H n \ er( j (T )) 1jn is invariant for T and T jh n is an n-isoetry. Proposition 1.6 in [3] states that if T 2 B(H) is an -isoetry, then H 1 is the axial invariant subspace M such that T jm is an ( 1)-isoetry. It is natural to as if H n in Proposition 3.1 is in fact the axial invariant subspace M such that T jm is an n-isoetry for n 1; 2; : : : ; 2: Next, we will identify soe other invariant subspaces. For 1; another subspace K is de ned by K (T ) K : \ i0 er( (T )T i ): (9) It follows fro (1) that K 1 K 2 K 3 : Siilarly we have the following result. 9

10 Proposition 3.3 Let T 2 B(H). Then K is invariant for T and T jk is an -isoetry. Proof. If x 2 K ; then (T )T i x 0 for all i 0: Thus (T )T i T x (T )T i+1 x 0 for all i 0: So T x 2 K and K is invariant for T: Furtherore (T jk ) P K (T )jk 0 because K er( (T )): In fact, there is a subspace K 1 de ned by K 1 (T ) K 1 : h 2 H : li sup (T )T i h 1 0 for all i 0 :!1 Siilarly, we de ne K 1 (T ) : h 2 H : li sup (T )T i h 1 0 for all i 0!1 and U 1 : K 1 (T ) T K 1 (T ): (10) Proposition 3.4 Let T 2 B(H). Then K 1 is invariant for T and T jk 1 is an 1-isoetry. Proof. By de nition, if x; y 2 K 1, then for i 0 and any two coplex nubers a and b; li sup (T )T i (ax + by) 1!1 li sup (jaj (T )T i x + jbj (T )T i x ) 1!1 li sup (jaj (T )T i x ) 1 + li sup (jbj (T )T i x ) 1 0:!1!1 Hence K 1 is a subspace. The proof that K 1 is invariant for T is also straightforward. To prove T jk 1 is an 1-isoetry, we need to show Note that (T jk 1 ) P K1 (T )jk 1 : Hence for h 2 K 1 ; li sup (T jk 1 ) 1 0: (11)!1 li sup (T jk 1 )h 1 li sup P K1 (T )h 1!1!1 li sup (T )h 1 0: (12)!1 We will use the unifor boundedness principle to prove (11). But a straightforward application of the unifor boundedness principle to the sequence of operators f (T jk 1 )g only gives (T jk 1 ) C for 1 and li sup (T jk 1 ) 1 1;!1 where C is soe constant. However, note that for any > 0 and h 2 K 1 ; by (12), li sup (T jk 1 )h 1 li sup (T jk 1 )h 1 0:!1!1 10

11 By applying the unifor boundedness principle to the sequence of operators f (T jk 1 )g ; we have (T jk 1 ) C (for 1) and where C is soe constant. Therefore (T jk 1 ) 1 C1 ; li sup (T jk 1 ) 1 C 1 li sup!1!1 1 : Since > 0 is arbitrary, this proves that li sup!1 (T jk 1 ) 1 0: Decoposition theores for operators are iportant in operator theory, see the boo [22]. The Nagy-Foias-Langer decoposition theore for a contraction says every contraction is a direct su of an unitary and a copletely nonunitary contraction, see Theore 5.1 in [22] for details. The following result generalizes Nagy-Foias-Langer decoposition theore in two ways. First, it is valid for an arbitrary operator. Second, it is valid for -unitaries for all 1 and for 1-unitaries. This also gives a di erent (probably ore direct) proof of Nagy- Foias-Langer decoposition theore since the proof of Nagy-Foias-Langer decoposition theore use the assuption of T being a contraction in an essential way. Theore 3.5 Let T 2 B(H): (a) Then U de ned by the forula U : K (T ) T K (T ) \ i0 er( (T )T i ) T er( (T )T i ) (13) is the unique axial reducing subspace on which T is an -unitary. Furtherore, T T 1 T 2 with respect to the decoposition H U U? ; where T 1 is an -unitary and T 2 is an operator which has no direct -unitary suand. (b) The space U 1 de ned by (10) is the unique axial reducing subspace on which T is an 1-unitary. Furtherore, T T 1 T 2 with respect to the decoposition H U 1 U? 1; where T 1 is an 1-unitary and T 2 is an operator which has no direct 1- unitary suand. Proof. We rst prove (a). By Proposition 3.3, K (T ) de ned in (9) is invariant for T and K (T ) is invariant for T ; thus U is reducing for T: It also follows that T ju is an -unitary. Now if M H is reducing for T and T jm is an -unitary, then (T )jm P M (T )jm (T jm) 0: So M er( (T )): Since M is invariant for T; T i M M er( (T )) for each i 0: This iplies that M er( (T )T i ): Hence M K (T ): Siilarly, since M H is reducing for T and T jm is an -unitary, we have M K (T ): Thus M U : This proves the axiality of U : The decoposition T T 1 T 2 now follows fro the axiality of U : 11

12 For (b), we only prove the axiality of U 1 since the rest of the proof is siilar to that of (a). Assue M H is reducing for T and T jm is an 1-unitary. Then (T )jm P M (T )jm (T jm): So, for any h 2 M and i 0 li sup (T )T i h 1 li sup (T jm)t i h 1!1!1 li sup (T jm) 1 T i h 1 0:!1 By the de nition of K 1 (T ); h 2 K 1 (T ): So, M K 1 (T ): Siilarly, since M H is reducing for T and T jm is an 1-unitary, we have M K 1 (T ): In conclusion M U 1 : The proof is coplete. The following de nition is natural. De nition 3.6 If T 2 B(H) is an -unitary for 2, T is called a pure -unitary if T has no nonzero direct suand which is an ( 1)-unitary. Siilarly, if T 2 B(H) is an 1-unitary, T is called a pure 1-unitary if T has no nonzero direct suand which is a nite-unitary. For each 1; let U denote the unique axial reducing subspace as in (13) such that T ju is an -unitary. We have the following decoposition theore for an 1-unitary. Theore 3.7 Let T 2 B(H) be an 1-unitary. Let V 1 U 1 ; V n U n U n 1 for n 2 and V 1 H _ fui ; i 1g H _ fvi ; i 1g : Then V i is reducing for T for each i 1; 2; : : : ; 1; and with respect to the decoposition H V 1 V 1 V 2 V 3 ; T has the following for T V 1 V 1 V 2 V n ; where T jv 1 is a pure 1-unitary, T jv 1 is an unitary, and T jv n is a pure n-unitary for n 2: For a 2-isoetry T; Proposition 1.25 in [3] identi es the unique axial reducing subspace R 1 such that T jr 1 is an isoetry. Here we are able to identify R 1 for an arbitrary operator. This also leads to a decoposition theore for T siilar to Nagy-Foias-Langer decoposition for a contraction where the unitary part is replaced by the isoetric part. Theore 3.8 Let T 2 B(H). Then R 1 de ned by the forula R 1 R 1 (T ) : \ er( 1 (T )T i T n ) (14) i;n0 is the unique axial reducing subspace on which T is an isoetry. Furtherore, T T 1 T 2 with respect to the decoposition H R 1 R? 1 ; where T 1 is an isoetry and T 2 is an operator which has no direct isoetry suand. 12

13 Proof. We rst prove that R 1 is reducing for T: Let h 2 R 1. That is, 1 (T )T i T n h 0 for all i; n 0: We need to show both T h and T h are in R 1 : Clearly 1 (T )T i T n T h 1 (T )T i T n+1 h 0 for all i; n 0: Hence T h 2 R 1 : Note also if n 0; then Furtherore, if n 1; then for all i 0; 1 (T )T i T n T h 1 (T )T i+1 h 0 for all i 0: 1 (T )T i T n T h 1 (T )T i T n 1 T T h 1 (T )T i T n 1 [ 1 (T ) + I] h 1 (T )T i T n 1 1 (T )h + 1 (T )T i T n 1 h : Hence, T h 2 R 1 : In conclusion, R 1 is reducing for T: Note that by (9), R 1 K 1. By Proposition 3.3, T jk 1 is an isoetry. Thus T 1 T jr 1 is an isoetry. Next, we prove the axiality of R 1. Suppose now M is reducing for T and T jm is an isoetry. Then, if h 2 M; 1 (T )T i h [P M 1 (T jm)] (T i h) (0) (T i h) 0 for all i 0: So, M er( 1 (T )T i ) and M K 1 as in (9). Since M is reducing for T; T n h 2 M K 1 for all n 0: Equivalently, 1 (T )T i T n h 1 (T )T i T n h 0 for all i; n 0: Therefore, h 2 R 1 and M R 1 : The proof is coplete. The axial reducing subspace in Proposition 1.25 in [3] for a 2-isoetry is de ned soewhat di erently. Of course, with soe wor, one can prove that R 1 de ned by (14) reduces to the one in [3] for a 2-isoetry. It sees rather iraculous the proof wored for R 1 since we are unable to prove the sae conclusion for the analogous R with > 1; where R : \ er( (T )T i T n ): i;n0 13

14 4 Quasinilpotent perturbations of 1-isoetries We will use the following Lea 1 and Lea 7 fro [19] and Lea 8 in [17]. Recall that T 1 and T 2 in B(H) are double couting if T 1 T 2 T 2 T 1 and T 1 T 2 T 2 T 1 : Lea 4.1 Assue T 1 and T 2 2 B(H) are double couting, and T and Q 2 B(H) are couting. Then (T + Q) (T + Q ) 1 Q 2 3 (T )T 2 Q 1 ; ; 2 ; 3 (T + Q) 1 ; 2 ; 3 ; T 1 Q ( 2+ 4 ) 3 (T )T 2 Q ( 1+ 4 ) ; (15) (T 1 T 2 ) T1 (T 1 )T1 (T 2 ): (16) 0 Theore 4.4 below has its inspiration fro sub-jordan operators in [1] and [20]. It is directly suggested by the following result for -isoetries. Theore 4.2 Assue T and Q 2 B(H) are couting, and also assue that T is an - isoetry and Q is a nilpotent operator of order n. Then T + Q is an ( + 2n 2)-isoetry. The above result was proved for 1 in [10]. The general and soeties slightly iproved versions were discussed independently in [7], [19] and [23]. We rst prove a lea. Lea 4.3 Assue T and Q 2 B(H) are couting. Then (T + Q) C ax n(t ) + ax nl nl Qn ; where l 3 is the integer part of and 3 Proof. By Lea 4.1, Let l 3 (T + Q) i C 2 [T + Q] T + 1 : , the integer part of and i l (T + Q ) 1 Q 2 3 (T )T 2 Q 1 : 1 ; 2 ; 3 : For i 1; 2; 3; we de ne (T + Q ) 1 Q 2 3 (T )T 2 Q 1 : 1 ; 2 ; 3 14

15 Since iplies that i l for one of the i 1; 2; 3; we have (T + Q) 1 ; 2 ; (T + Q ) 1 Q 2 3 (T )T 2 Q 1 (17) : We now estiate i : We will prove the estiate for 3 since the proofs of the estiates for 1 and 2 are siilar. Note that 3 (T + Q ) 1 Q 2 3 (T )T 2 Q 1 1 ; 2 ; and 3 l 1 ; 2 ; and 3 l (T + Q ) 1 Q 2 3 (T ) T 2 Q 1 ax n(t ) nl 1 ; 2 ; (T + Q ) 1 Q 2 T 2 Q 1 ax nl n(t ) ([T + Q ] Q + Q T + 1) (C2) ax nl n(t ) ; where the last equality follows fro ultinoial forula. Siilarly, by noting that we have Therefore by (17), The proof is coplete. (T ) (T + 1) for 1; 1 ax nl Qn ([T + Q ] + Q T + [T + 1]) ax nl Qn (C2) ; 2 ax nl Qn " l ([T + Q ] Q + T + [T + 1]) ax nl Qn (C2) : (T + Q) (C2) ax n(t ) + 2(C2) ax nl nl Qn C ax n(t ) + ax nl nl Qn : 15

16 Theore 4.4 Assue T and Q 2 B(H) are couting, and also assue that T is an 1-isoetry and Q is a quasinilpotent operator. Then T + Q is an 1-isoetry. Proof. Given 1 > " > 0; let N be such that n (T ) " n ; Q n " n for n N: Then by Lea 4.3, for 3N; we have l 3 N and (T + Q) C ax nl n(t ) + ax nl Qn 2C " [ 3 ] for soe constant C: Hence li sup!1 (T + Q) 1 0 and T + Q is an 1-isoetry. In view of Theore 4.2 and Theore 4.4, we ae the following conjecture. Conjecture 4.5 Assue T and Q 2 B(H) are couting, and also assue that T is an -isoetry for soe 1 and Q is a quasinilpotent operator but not a nilpotent operator. Then T + Q is an 1-isoetry but not a nite-isoetry: We now prove a liit result alluded in the introduction. Proposition 4.6 If T n 2 B(H) is a sequence of couting 1-isoetries and T n! T in operator nor, then T is an 1-isoetry. Proof. By the assuption T n T l T l T n for all n; l 1; we have T T n T n T for all n 1: The proof is siilar to the previous proof. Given 1 > " > 0; let N be such that T T N " and n (T N ) " n for n N: Then by Lea 4.3, for 3N; l 3 N and (T ) (T N + T T N ) C ax (T N ) + ax (T [3] [3] C ax (T N ) + ax T [3] [3] 2C " [ 3 ] TN ) T N for soe constant C: Hence li sup!1 (T ) 1 0 and T is an 1-isoetry. Proble 4.7 What happens when T n are not couting in Proposition 4.6? 16

17 There is a partial converse to Theore 4.4. Let K be another coplex Hilbert space. Let H K denote the Hilbert space tensor product of H and K: Let I H and I K denote the identity operators on H and K respectively. If T 2 B(H) is an 1-isoetry and S 2 B(K) is a quasinilpotent operators, then T I K on H K is an 1-isoetry and T 2 I H S is a quasinilpotent operators. Thus, by Theore 4.4, T I K + I H S is an 1-isoetry on H K: We will prove the converse of this result under a technical assuption. We rst prove a lea. Lea 4.8 Let T 2 B(H) and Q 2 B(K): If (T I H + I K Q) is an 1-isoetry on H K and 0 2 ap (Q); then T is an 1-isoetry. Proof. By assuption, 0 2 ap (Q): That is, there exist unit vectors i 2 K such that Q i! 0 as i! 1: Thus, for any l > 0; Q l i! 0 as i! 1: Note forula (15) in Lea 4.1 becoes (S I + I Q) T 1 3 (T )T 2 Q ( 2+ 4 ) Q ( 1+ 4 ) : 1 ; 2 ; 3 ; So, for any unit vector h 2 H; letting i! 1; we have h (S I + I Q)(h i ); (h i )i ; 2 ; 3 ; 4 ht 1 3 (T )T 2 h; hi Q ( 1+ 4 ) i ; Q ( 2+ 4 ) i! h (T )h; hi ; since all the ters in the above suation tend to zero except when and 3 : Note that (h i ) is a unit vector in H K: Hence (S I + I Q) jh (S I + I Q)(h i ); (h i )ij and (S I + I Q) jh (T )h; hij : Since the unit vector h 2 H is arbitrary, (S I + I Q) is bigger than or equal to the nuerical radius of (T ): But (T ) is a self-adjoint operator and thus its nuerical radius and the nor are the sae. Therefore (S I + I Q) (T ) : This iplies that if li sup!1 (S I + I Q) 1 0, then li sup!1 (T ) 1 0: So T is an 1-isoetry on H: Note that for any constant ; (T + I H ) I K + I H (Q I K ) S I + I Q: By a translation we can assue 0 2 ap (Q): So a little re ection gives the following interesting corollary. 17

18 Corollary 4.9 Let T 2 B(H) and Q 2 B(K): Then (S I + I Q) ax f (T + I H ) : 2 ap (Q)g ; (S I + I Q) ax f (Q + I K ) : 2 ap (T )g : The following technical condition is siilar to the one needed for an analogous result for -isoetries (Theore 12 in [17]). Let be the set of four points on the unit speci cally, e i e i for soe ; 2 [0; 2): (18) Theore 4.10 Let T 2 B(H) and Q 2 B(K): Assue (T I H + I K Q) 6 : Then (T I H + I K Q) is an 1-isoetry on H K if and only if one of the following holds. I is a quasi- (a) There exists a constant such that T + I is an 1-isoetry and Q nilpotent operator. (b) There exists a constant such that Q + I is an 1-isoetry and T nilpotent operator. I is a quasi- Proof. Assue (T I H + I K Q) is an 1-isoetry. We will prove either (a) or (b) holds. The proof is siilar to and slightly sipler than the proof of Theore 12 in [17]. We rst prove that either (T ) or (Q) is a singleton or both (T ) and (Q) contain exactly two points. If (Q) is not a singleton, let 1 ; 2 be two di erent nubers in ap (Q): We use I to denote either I H or I K : Since it follows that 0 2 ap (Q that (T + i I H ) I K + I H (Q i I K ) S I + I Q; i 1; 2; i I) : By Lea 4.8, T + i I is an 1-isoetry for i 1; 2: Note (T + 2 I) (T + 1 I) : But by Proposition 2.1, either (T + 1 I) D or (T + 1 If (T + 1 I) D ; then T + 2 I can not be an 1-isoetry since a translation of the unit dis D is not the unit dis. Therefore (T + 1 It is clear that (T + 2 I) 6 D, so (T + 2 as well. But (T + 2 I) is a translation of (T + 1 I) by the nuber 2 1 ; thus (T 1 I) consists of at ost two points. In the case (T + 1 I) is a singleton we are done. In the case (T ) consists of two points 1 ; 2 ; since 1 ; 2 2 ap (T ), a siilar arguent shows that (Q 1 I) consists of exactly two points as well. We rst deal with the singleton case. Without loss of generality, let (Q) fg : Then Q I is a quasinilpotent operator. By Lea 4.8, T + I is an 1-isoetry. That is, (b) holds. In this case, both (T ) and (Q) contain exactly two points. Since (e i (T + I) I + I e i (Q I)) e i (T I + I Q) ; 18

19 by a rotation and a translation, we ay assue that (Q) ap (Q) f0; g for soe > 0: By a theore of Rosenblu [29], (T I + I Q) (T ) + (Q): But by Proposition 2.1, (T I + I Hence (T \ ( and (T ) contains exactly two points when 0 < < 2. In fact, we have Therefore (T ) ap (T ) e i ; e i ; where e i r1 2 + i (T I + I Q) (T ) + (Q) e i ; e i ; e i ; e i e i ; 2 4 : which can not happen by our assuption. The proof is coplete. See Proposition 14 in [17] for exaples where it is shown that the corresponding result for -isoetries ay not hold without the assuption (T I H + I K Q) 6 : The following result gives a partial answer to Conjecture 4.5. Proposition 4.11 Let T 2 B(H) and Q 2 B(K): Assue (T I H + I K Q) 6 : If T is an -isoetry for soe 1 and Q is a quasinilpotent operator but not a nilpotent operator. Then (T I H + I K Q) is an 1-isoetry but not a nite-isoetry. Proof. It follows fro Theore 4.4 that (T I H + I K Q) is an 1-isoetry. We prove by using contradiction. Assue (T I H + I K Q) is a strict n-isoetry for soe n 1: By Theore 12 in [17], either there exists a constant such that T + I is a -isoetry and Q I is a nilpotent operator of order l with + 2l 2 n or there exists a constant such that Q + I is an -isoetry and T I is a nilpotent operator of order l. In the rst case 0; and Q is a nilpotent operator of order l; which contradicts the assuption on Q. In the second case, Q + I is a -isoetry iplies that jj 1: So (Q + I) fg : Recall that if S is an -isoetry for soe 1 and (S) consists of a nite nuber of points, then S is the direct su of operators of the for I + Q 0 ; where Q 0 is a nilpotent operator, see Proposition 11 in [17]. Hence, Q + I is a -isoetry, which iplies that Q is a nilpotent operator. This again contradicts the assuption on Q. 5 Tensor products of 1-isoetries The following result is inspired by the wor on products of -isoetries in [9]. In fact, a result is proved there for products of -isoetries on Banach spaces. See also [19] and [23] for related results for -isoetries on Hilbert spaces. 19

20 Theore 5.1 Assue T 1 are T 2 in B(H) are double couting. If T 1 and T 2 are 1- isoetries, then so is T 1 T 2. Proof. By Lea 4.1, (T 1 T 2 ) 0 T1 (T 1 )T1 (T 2 ): This proof is siilar to the proof of Lea 4.3 and Theore 4.4 by using the above forula. The proof here is actually slightly sipler. For clarity, we include the proof. Given 1 > " > 0; let N be such that n (T 1 ) " n and n (T 2 ) " n for n N: We clai that there exists a constant C such that for 2N Let l 2 (T 1 T 2 ) C " 2 : denote the integer part of : We write 2 (T 1 T 2 ) I + II; where I : II : l 0 l+1 T 1 (T 1 )T 1 (T 2 ); T 1 (T 1 )T 1 (T 2 ): Note that for l 2 ; 2 l N; so Note also by de nition Therefore I " l (T 1 ) " " l : (T 2 ) (T 2 + 1) for 1: l l T 1 T 1 " (T 1 ) T 1 (T 2 ) T 1 (T 2 + 1) T 1 T 1 (T 2 + 1) " l T 1 2 (T 2 + 1) + 1 : 20

21 Siilarly, by noting that for l + 1 N; (T 2 ) " ; we have In conclusion, for n 2N II " l T (T 1 + 1) : (T 1 T 2 ) " [ 2 ] T1 2 (T 2 + 1) T1 2 + (T 1 + 1) : Therefore li sup!1 (T 1 T 2 ) 1 0 and T 1 T 2 is an 1-isoetry. By applying Theore 5.1, we have the following result: if T 2 B(H) and S 2 B(K) are 1-isoetries, then T S on H K is an 1-isoetry. To see this, let T 1 T I K and T 2 I H S; where I H and I K are identity operators. Then T 1 and T 2 are double couting, and T 1 and T 2 are 1-isoetries. Thus by Theore 5.1, T 1 T 2 T S is an 1-isoetry. Corollary 5.2 If T 2 B(H) and S 2 B(K) are 1-isoetries, then T S on H K is an 1-isoetry. A siilar result on tensor products of -isoetries is the following: Theore 5.3 [16] If T 2 B(H) is an -isoetry and S 2 B(K) is an n-isoetry, then T S on H K is an ( + n 1)-isoetry. We would lie to ention that the result in Theore 5.3 on tensor products of - isoetries was rst forulated in ter of eleentary operators on the ideal of Hilbert- Schidt operators in [11] and [12]. This result was proved in Theore 2.10 of [16]. Siple proofs of slightly iproved versions of this result were given in [19] and [23]. A converse to Theore 5.3 was obtained in Theore 7 of [17]. We now prove an analogous result of Theore 7 of [17] for tensor products of 1- isoetries. Theore 5.4 Let T 2 B(H) and S 2 B(K): Then T S on H K is an 1-isoetry if and only if both T and S are 1-isoetries on H and K respectively for soe constant. Proof. Note that for T 1 T I K and T 2 I H S; forula (16) becoes (T S) 0 T (T )T (S): One way of the theore is Corollary 5.2 by noting that (T ) (S) T S. Now assue T S is an 1-isoetry. By Proposition 2.1, either (T or (T S) D : By a theore fro Brown and Pearcy [13], (T S) (T ) (S): In particular 1 r(t S) r(t ) r(s): If 1 ; then r(t ) r( 1 S) 1: Since r(t ) 21

22 T 1 S T S; we ay assue r(t ) r(s) 1: We will prove that S is an 1-isoetry and the proof for T is siilar. By the assuption r(t ) 1; there is a such that 2 ap (T ): Let h i be a sequence of unit vectors in H such that (T )h i! 0: Then for any 0; T h i 2! jj 2 1 as i! 1: Thus as in the proof of Proposition 2.1, for each j > 0; 0; j j T j (T )T h i ; T h i ( 1) j l l T h i 2 l l0 j j! ( 1) j l 0 as i! 1: (19) l l0 Since by (19), (T )T h i ; T h i! 0 for 6 ; therefore for any unit vector x 2 K, Thus h (T S)(h i x); h i xi T (T )T h i (S)x; h i x 0 (T )T h i ; T h i h (S)x; xi 0! h (S)x; xi as i! 1: (T S) jh (T S)(h i x); h i xij and (T S) jh (S)x; xij : Since (S) is a self-adjoint operator, we have (T S) (S) : This iplies that if li sup!1 (T S) 1 0; then li sup!1 (S) 1 0: So, S is an 1-isoetry on H. A little re ection yields the following interesting corollary. Corollary 5.5 Let T 2 B(H) and S 2 B(K): Then for 1 (T S) (r(t )S) and (T S) (r(s)t ) : We rear that several previous results for 1-isoetries also have corresponding results for 1-unitaries. We state one of the. Theore 5.6 Let T 2 B(H) and S 2 B(K): Then T S on H K is an 1-unitary if and only if both T and S are 1-unitary on H and K respectively for soe constant. Acnowledgeent. The research of Muneo Chō is partially supported by Grant-in-Aid Scienti c Research No.15K The wor of Woo Young Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea governent (MSIP) (No ). 22

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