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Square roots of some classical operators

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Ces\`{a}ro operator.


INTRODUCTION
If H is a complex Hilbert space and A ∈ B(H), the bounded linear operators on H, when does A have a square root? By this we mean, does there exist a B ∈ B(H) such that B 2 = A? If A has a square root, can we describe {B ∈ B(H) : B 2 = A}, the set of all the square roots of A?
First let us make the, perhaps unexpected, observation that not every operator has a square root. For example, Halmos showed that the unilateral shift Sf = zf on the Hardy space H 2 [4] does not have a square root [6]. Other examples of operators constructed with the shift S and its adjoint S * , for example S ⊕ S * and S ⊗ S * , also do not have square roots [2]. See the papers [8,12,13,18,19] for some general results concerning square roots of operators.
Second, many operators have an abundance of square roots. For example, any nilpotent operator of order two is a square root of the zero operator. Moreover, to highlight their abundance, Lebow proved (see [7,Prob. 111]) that when dim H = ∞, the set {A ∈ B(H) : A 2 = 0} is dense in B(H) in the strong operator topology.
Much of the work on square roots has focused on the general topic of which operators have square roots and the prevalence of types of square roots (pth roots and logarithms) in B(H). Previous papers also have results which explore the relationship between the type of square root as related to the type of operator. In this paper, we focus on a collection of some well-known classical operators and proceed to characterize all of their square roots. The classical operators included in this discussion are the square of the unilateral shift (Theorem 2.4), the Volterra operator (Theorem 3.7), certain compressed shifts (Theorem 4.1), the unilateral shift plus its adjoint (Theorem 5.2), the Hilbert matrix (Theorem 6.2), and the Cesàro operator (Theorem 7.2 and Theorem 7.4). Our work on the Cesàro operator answers a question posed in [12] and stems from a question posed by Halmos.

SQUARE ROOTS OF S 2
Suppose that (Sf )(z) = zf (z) denotes the unilateral shift on the Hardy space H 2 [11]. In this section we explore the square roots of S 2 . One square root of S 2 is, of course, S itself. Our characterization of all of the square roots of S 2 , requires a few preliminaries.

Our last bit of notation is the vector-valued shift
It is traditional to think of S ⊕ S in matrix form as The above formulas yield the following well-known fact. Some other well-known facts used in the section involve the commutants of S and S ⊕ S. For ϕ ∈ H ∞ , the bounded analytic functions on D, the Toeplitz (Laurent) operator T ϕ g = ϕg is bounded on H 2 and ST ϕ = T ϕ S.
In a similar way, let where Φ ij ∈ H ∞ . We use the notation M 2 (H ∞ ) for the 2 × 2 matrices above with H ∞ entries. Define T Φ : A calculation shows that (S ⊕ S)T Φ = T Φ (S ⊕ S). Similar to the above, we have the following [20,Cor. 3.20].
Here is the main theorem of this section describing all of the square roots of S 2 .
Theorem 2.4. For Q ∈ B(H 2 ) the following are equivalent.
(ii) There is a 2 × 2 constant unitary matrix U and functions a, b, c ∈ H ∞ satisfying (ii) There is a 2 × 2 constant unitary matrix U and functions a, b, c ∈ H ∞ satisfying A matrix calculation shows that In the above, we use the fact that any constant matrix commutes with S ⊕S. Thus every operator of the form (2.9) is a square root of S ⊕ S. The rest of this section will be devoted to proving the converse -and providing some instances of this characterization.
Our proof involves a few more preliminaries. The first is a simple fact about square roots of bounded Hilbert space operators. Proof. Note that AB = AA 2 = A 2 A = BA.
Combining this with the discussion above, we see that if Q ∈ B(H 2 ) with Q 2 = S 2 , then W QW * ∈ B(H 2 ⊕ H 2 ) with (W QW * ) 2 = S ⊕ S. It follows from Lemma 2.10 and Proposition 2.3 that To identify A, let us start with lemmas about 2 × 2 matrices One can quickly verify the following useful facts about the subspace Lemma 2.12. Let α, β, γ, λ ∈ C.
Each term A k (S ⊕S) k belongs to B(H 2 ⊕H 2 ) as does each partial sum of the series above. If we suppose that the series above converges in the strong operator topology, then A ∈ B(H 2 ⊕ H 2 ). Suppose U ∈ M 2 (C) is a constant unitary matrix. A simple 2 × 2 matrix calculation shows that This yields the important identity Proof of Theorem 2.4. We will prove Theorem 2.7. Proposition 2.1 will then imply Theorem 2.4.
where a, b, c, d ∈ H ∞ . Let a k , b k , c k , d k denote the Taylor coefficients of a, b, c, d respectively and define For the matrix A 0 , Schur's theorem provides us with a unitary matrix U such that U A 0 U * is upper triangular. By (2.13) (and unitary equivalence) we can always assume that A is a square root of (S ⊕ S) with A 0 being upper triangular. Since Comparing the operator coefficients in front of each (S ⊕ S) k we have Now we will inductively find a formula for A.
The matrix A 0 is upper triangular. By (2.14) and Lemma 2.12, By (2.15) and Lemma 2.12 we get b 0 = 0 and for arbitrary a 1 , b 1 .
We will now use induction to prove that A k ∈ S. The base cases A 0 , A 1 belong to S. By Lemma 2.12, right hand side of (2.16) or (2.17) are constant multiples of the identity operator I on H 2 ⊕ H 2 . Thus, by Lemma 2.12, A k ∈ S.

By the expansion
it follows that a 2 + bc = z. Equivalently, by relabeling a, b, c, we can write where a, b, c ∈ H ∞ with za 2 + bc = 1.
The converse was shown earlier.
Remark 2.18. (i) Since unitary operators preserve determinants, every square root A of S ⊕ S will satisfy det A = −z.
(ii) It follows from Proposition 2.3 and Proposition 2.1 that every B ∈ {S 2 } is of the form (Bg)(z) = ϕ(z)g e (z)+ψ(z)g o (z) for some ϕ, ψ ∈ H ∞ . This is an interesting (and known) fact.
(iv) Taking in Theorem 2.4, yields another class of square roots Q of S 2 : With respect to the standard basis, Q has the matrix representation, which is just the "obvious" square root of S 2 , namely S.
This brings us to a brief comment as to when the square root of S 2 is a (analytic) Toeplitz operator. Here is a general fact concerning Toeplitz op- [4,Ch. 4] for the basics of Toeplitz operators on H 2 .
(i) There is a Toeplitz operator T such that T 2 = T ϕ .
Below we will use the tensor product x⊗y of two vectors in a Hilbert space H. This is the rank-one operator on H defined by (x ⊗ y)(z) = z, y x. Recall that I = SS * + 1 ⊗ 1 and thus Therefore, either S * T ψ 1 = 0 or S * Tψ1 = 0. These identities respectively mean P + ψ or P +ψ are constant functions. We can rephrase these conditions as ψ ∈ H ∞ or ψ ∈ H ∞ . However, under any of these two conditions, which implies ϕ = ψ 2 . In the former case, when ψ ∈ H ∞ , we may replace ψ byψ so that always ψ is an analytic function and then either ϕ = ψ 2 or ϕ = ψ 2 .
The previous theorem, along with the standard inner-outer factorization of H ∞ functions yields the following corollary. We end this section with the remark that S 2n has infinitely many square roots since S 2n is unitarily equivalent to (S ⊕ S) (n) , and we already know that S ⊕ S has infinitely many square roots. However S 2n+1 does not have any square roots. We will discuss these results and some others in a forthcoming paper.

SQUARE ROOTS OF THE VOLTERRA OPERATOR
The Volterra operator is a well-known bounded operator on L 2 [0, 1] with a known square root [20, p. 81] One can prove this using the Laplace transform and convolutions. For the sake of completeness, and since the ideas of the proof will be used in the next section, we give an exposition of the following result of Sarason from [24]. Our presentation will be somewhat different from Sarason.
Theorem 3.2 (Sarason). The operators ±Y are the only two square roots of V .
If Θ is the atomic inner function a result of Sarason [23] (see also [20,Ch. 4]), shows that for g ∈ L 2 [0, 1], the function belongs to the model space K Θ = H 2 ∩ (ΘH 2 ) ⊥ and the operator J : The same paper says that For ψ ∈ H ∞ , the operator ψ(S Θ ) can be written as a truncated Toeplitz operator. Indeed, for any ψ ∈ L ∞ (T), define the operator A ψ on K Θ by via radial boundary values, as a subspace of L 2 (T)).
Let us record some facts about truncated Toeplitz operators that will be used below. One can find their proofs in [4] or [24].
Though the operator (I − S Θ )(I + S Θ ) −1 is well defined, we need to represent it as a truncated Toeplitz operator with an H ∞ symbol. This is accomplished with the following.
Recall that ϕ is an outer function (and hence is zero free in D) and so there are indeed ψ ∈ H ∞ with ψ 2 = ϕ. This says that A = J * A ψ J for some ψ ∈ H ∞ with ψ 2 = ϕ + Θh for some h ∈ H ∞ .
On the other hand, if ψ ∈ H ∞ and h ∈ H ∞ with ψ 2 = ϕ + Θh, then the operator Note the use of the H ∞ functional calculus for the compressed shift S Θ as well as the fact that A Θ = 0 (Proposition 3.4). This argument is summarized with the following theorem. (ii) A = J * A ψ J for some ψ ∈ H ∞ such that ψ 2 = ϕ + Θh for some h ∈ H ∞ .
To show there are only two square roots of V , we follow a variation of an argument of Sarason [24]. Notice that one square root of V is J * A √ ϕ J. Let us show that the other is J * A − √ ϕ J. If B is another square root of V , then B = J * A ψ J where ψ 2 = ϕ + Θh. In other words, ψ 2 − ϕ = Θh. Write and observe that for some γ j 0, the inner functions Moreover, choose the largest γ 1 , γ 2 such that q 1 and q 2 are inner divisors of ψ − √ ϕ and ψ + √ ϕ. Write It follows that √ ϕ = 1 2 (q 1 h 1 − q 2 h 2 ). Since √ ϕ is outer, it must be the case that one of γ 1 or γ 2 must be zero. If γ 1 > 0 and γ 2 = 0, then γ 1 1 and it follows that ψ + √ ϕ is divisible by Θ. An application of Proposition 3.4 Notice how the closure of this image does not intersect the origin.

SQUARE ROOT OF A COMPRESSED SHIFT
The previous section leads us to a discussion about the square roots of a compressed shift. For any inner function u, there is the compressed shift S u = P u S| Ku . The proof of Theorem 3.7 implies the following theorem.
Theorem 4.1. For the atomic inner function Θ and A ∈ B(K Θ ), the following are equivalent.
Furthermore, S Θ has exactly two square roots.
Proof. First let us prove that the set of square roots of S u is nonempty. For this it is enough to check that z + Θ(z)(1 − z) 1/5 has no zeros in D (see Figure 2) and thus has an analytic square root. The reasoning is similar to the argument in Proposition 3.5, albeit a bit more complex. In this case and thus f (e iθ ) = cos θ + (2 sin θ/2) 1/5 cos θ−π 10 − cot(θ/2) , 0 < θ < π.
There is a similar formula for −π < θ < 0. Then it is enough to observe that By the Poisson integral formula, we conclude that f (z) m for all z ∈ D.
Thus f is outer and hence has no zeros in D.
Next we observe that A √ f is a square root of S Θ . Now follow the argument used to prove there are only two square roots of the Volterra operator (following Theorem 3.7) to prove that the other square root of S Θ is A − √ f . Not every compressed shift has a square root.

Proposition 4.2.
Suppose u is inner and u has a zero at z = 0 of order at least two. Then S u does not have a square root.
Proof. Our earlier discussion shows that the set of square roots of S u are {A ψ : ψ ∈ H ∞ , ψ 2 = z + uh, h ∈ H ∞ }. If u has a zero of order at least two at z = 0, then z + uh = z + z 2 k for some k ∈ H ∞ and thus z + z 2 k(z) has a zero of order one at z = 0. Thus, there is no H ∞ function ψ for which ψ 2 (z) = z + uh.

SQUARE ROOTS OF T cos θ .
The Toeplitz operator with symbol cos θ, equivalently T cos θ = 1 2 (S + S * ), is a self-adjoint operator. Therefore, by the spectral theorem for normal operators, it has a square root.
A result of Hilbert [10] (see [22,Ch. 3] for a modern presentation) shows that if (u n ) ∞ n=0 are the Chebyschev polynomials of the second kind [25], then the operator F : is unitary and intertwines M x on L 2 (ρ) and T cos θ . More explicitly, Thus the matrix representation for T cos θ with respect to the orthonormal basis (z n ) ∞ n=0 for H 2 is [a mn ] ∞ m,n=0 , where a mn := T cos θ z n , z m H 2 , m, n 0, which is the Toeplitz matrix By means of the unitary operator F , we can also write a mn = 2 π Therefore, F M ϕ F * is a square root of T cos θ . Of course, there are many other ϕ for which ϕ 2 = x. The matrix representations of M ϕ (with respect to the Chebyschev basis) and F M ϕ F * (with respect to the standard basis for . In fact, these are all the square roots of A. (ii) With respect to the orthonormal basis (z n ) ∞ n=0 of H 2 , the matrix representation of B is Proof. Let B be any fixed square root of T cos θ . Since T cos θ is unitarily equivalent to M x on L 2 (ρ) via the unitary operator F defined by (5.1), the operator F * BF is a square root of M x . By Lemma 2.10, F * BF commutes with M x and thus equal to M ϕ for some ϕ ∈ L ∞ (ρ) (To see why this is true note, that if AM x = M x A, then Ap = pA(1) for any polynomial p. Letting f ∈ L 2 (ρ) and selecting polynomials p n → f in norm and, passing to a subsequence, almost everywhere, we see that Af = (A1)f . This shows that A1 ∈ L ∞ (ρ) and A is the operator multiplication by A1). This immediately implies By the uniqueness of the symbol of a multiplication operator, we must have ϕ(x) 2 = x. The matrix representation of F * BF = M ϕ with respect to the orthonormal basis ( 2 π u n ) ∞ n=0 of L 2 (ρ) is the same as the matrix representation of B with respect to the orthonormal basis (z n ) ∞ n=0 of H 2 , and is given by (5.3).
Notice that all of these square roots are complex symmetric operators, since with respect to the Chebyschev basis, the matrix representation (5.3) is self transpose.

SQUARE ROOTS OF THE HILBERT MATRIX
The square root of the Hilbert matrix as an operator on 2 , involves a similar analysis as with the Toeplitz matrix T cos θ from the previous section. But here, the spectral representation theorem of Hilbert is replaced by one of Rosenblum [21]. We outline this analysis here.
Lebedev [16,17] proved that if the modified Bessel function of the third kind, then the operator is a unitary operator from L 2 ((0, ∞), dx) to itself. Thus is an orthonormal basis for L 2 ((0, ∞), dx). Rosenblum [21] proves that if This last quantity equals He m , e n 2 (the entries of the Hilbert matrix). In summary, the linear transformation W : 2 → L 2 ((0, ∞), dx) defined by As in the previous section, if g ∈ L ∞ ((0, ∞), dx) with g 2 = h, then M g is a square root of M h and thus W * M g W is a square root of H. Conversely, if T ∈ B( 2 ) with T 2 = H, then W T W * is a square root of M h and hence, as we have seen several times before, W T W * belongs to the commutant of M h . Since h is a monotone decreasing function on (0, ∞), h is injective and hence by a well-known fact about multiplication operators, M h is cyclic.
Since the commutant of a cyclic multiplication operator is the set of multiplication operators M g on L 2 ((0, ∞), dx) with g ∈ L ∞ ((0, ∞), dx) we see as before that T = W * M g W , where g 2 = h. We therefore arrive at the following theorem. Below, we regard any T ∈ B( 2 ) as an infinite matrix.
Theorem 6.2. For T ∈ B( 2 ) the following are equivalent.
(ii) There is a measurable function g on (0, ∞) with g 2 = h, where h is the function from (6.1), such that .

SQUARE ROOTS OF THE CESÀRO OPERATOR
The Cesàro operator C : Some basic facts about C are found in [1]. With resect to the standard orthonormal basis (z n ) ∞ n=0 for H 2 , the matrix representation of C is which is known as the Cesàro matrix. Though not quite obvious, C has a square root and in fact, one can write them all down -since there are only two of them. This is the topic of this section. It is important to note here that by the Conway-Olin functional calculus for subnormal operators [3], one can prove that C has at least one square root. The purpose here is to show that C has exactly two square roots and to specifically write them down.
Our path to identify the square roots of C is through subnormal operators and the work of Kriete and Trutt. Along the way to proving that C is subnormal, a paper of Kriete and Trutt [14] shows that for w ∈ D, the function belongs to H 2 and satisfies (I − C * )v w = wv w . The space defines a vector space of analytic functions on D that becomes a Hilbert space, in fact a reproducing kernel Hilbert space, when endowed with the norm F H = f H 2 . This makes the operator (U f )(z) = F (z) a unitary operator from H 2 to H. Furthermore, In summary, C is unitarily equivalent to M 1−z on H.
Thus if A is a square root of C, then, as we have seen with the other operators covered in this paper, A ∈ {C} and thus U AU * ∈ {M 1−z } = {M z } . So now we need to identify {M z } . This will involve the multiplier algebra of H.
Another paper of Kriete and Trutt [15] argues that each ϕ ∈ H ∞ is a multiplier of H (i.e., ϕH ⊆ H). This is significant since for a general Hilbert space of analytic functions (the Dirichlet space for example), not every H ∞ function is a multiplier. Furthermore, a standard fact that the multiplier algebra of any reproducing kernel Hilbert space of analytic functions on D is contained in H ∞ , along with the observation above, shows that the multiplier algebra of H is H ∞ .
The Hilbert space H also contains the polynomials as a dense set and a standard argument, along with the discussion in the previous paragraph, Putting this all together, it follows that if A is a square root of C, then U AU * = M ϕ for some ϕ ∈ H ∞ . But and thus ϕ 2 = 1 − z on D. But since ϕ is analytic on D, it must be the case that ϕ(z) = ± √ 1 − z. Thus, the Cesàro operator has U * M √ 1−z U and U * M − √ 1−z U as its only square roots.
The above formulas for the square roots of C are a bit unsatisfying since they are hidden behind a unitary operator. Our goal in the next two results is to produce a more tangible description of the two square roots of C. Note that where the branch of the square root is taken so that √ 1 = 1. It is wellknown that (i) A 2 = C. (ii) converges in operator norm to M √ 1−z . But since M k z is unitarily equivalent to (I − C) k , we get The other square root of C is computed in a similar way.
Using an idea of Hausdorff [9], the paper [12] produces all of the lower triangular square roots of the Cesàro matrix from (7.1). That paper considers the Cesàro matrix and its resulting square roots as linear transformations on all one-sided sequences (not necessarily 2 sequences nor any assumption on the linear transformation being bounded). They show that all of the lower triangular square roots of the Cesàro matrix are the matrices where the sign along the diagonal is determined by the function σ.
They also conjecture that the choices of A σ , where σ ≡ 1 or σ ≡ −1, are the two bounded square roots of the Cesaro matrix (viewed as an operator on 2 ). This next theorem verifies this conjecture (thus answering a question posted by Halmos) and also gives an exact description of the square roots from Theorem 7.2.
Theorem 7.4. The following are equivalent for A ∈ B(H 2 ).
(ii) With respect to the orthonormal basis (z n ) ∞ n=0 for H 2 , the matrix representa- Proof. By the above discussion of the results from [12], all of the lower triangular square roots of the Cesàro matrix (as viewed as an operator on the space of all sequences) are of the form A σ for some σ : N → {−1, 1}. From (7.3), notice that and so the choice of σ is determined by the entries of A σ on its diagonal. If one of the bounded square roots of the Cesàro matrix from Theorem 7.2, notice that (I − C) k is lower triangular for all k 0 and thus so is A. We just need to determine which choice of σ yields A σ = A.
The (n, n) entry of I − C is (1 − 1 n+1 ) for n 0 and since I − C is lower triangular, it follows that the (n, n) entry of (I − C) k is (1 − 1 n+1 ) k . Thus, the (n, n) entry of A is But the above is just the Taylor series of and this turns out to be 1 n+1 . By (7.5), this corresponds to A σ with σ ≡ 1. When a similar analysis shows that corresponds to A σ with σ ≡ −1.
Thus the only two bounded square roots of the Cesàro (matrix) operator are  The first matrix in the sum above is one of the bounded square roots of the Cesàro operator while the second matrix in the sum turns out to be       1 0 0 0 0 · · · 1 0 0 0 0 · · · 1 0 0 0 0 · · · 1 0 0 0 0 · · ·