N-WEAKLY SUPERCYCLIC MATRICES

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1 N-WEAKLY SUPERCYCLIC MATRICES NATHAN S. FELDMAN Abstract. We define an operator to n-weakly hypercyclic if it has an orbit that has a dense projection onto every n-dimensional subspace. Similarly, an operator is n-weakly supercyclic if it has a scaled orbit that has a dense projection onto every n-dimensional subspace. In this paper, we show the following results: (i) There are no n-weakly hypercyclic matrices on R n or C n. (ii) There are no 2-weakly supercyclic matrices on C n for n 2. (iii) There are no 3-weakly supercyclic matrices on R n for n 3; and (iv) there are 2-weakly supercyclic matrices on R n if and only if n is even. Finally, we show that there is an onto isometry on l 2 R (N) that is 2-weakly supercyclic, but not 3-weakly supercyclic and also give some examples involving tuples of matrices. We conclude with some questions. 1. Introduction If T is a continuous linear operator on a Hilbert space H and x H, then the orbit of x under T is the sequence Orb(x, T ) = {T n x} n=0. If there exists a vector x H such that Orb(x, T ) is dense in H, then T is said to be hypercyclic and x is called a hypercyclic vector for T. The scaled orbit of x under T is F Orb(x, T ) = {ct n x : n 0, c F}, where F = R or C depending on whether H is a real or complex Hilbert space. The operator T is supercyclic if there is a vector in H whose scaled orbit is dense in H. For more on hypercyclicity and supercyclicity see the nice book by Bayart and Matheron [3]. It is well known that there are no hypercyclic matrices on F n for n 1 and that the only supercyclic matrices are rotation matrices on R 2 that rotate by an irrational multiple of π. In fact the results of this paper improve upon these facts. For in this paper we introduce a weaker form of hypercyclicity and supercyclicity and show that there are no matrices with the weak form of hypercyclicity, but that there are matrices with the weaker form of supercyclicity. For a real or complex Hilbert space H and an integer n 1, we define a set E H to be n-weakly dense in H if for every subspace M of H with dimension equal to n, the orthogonal projection of E onto M is dense in M. If T is a continuous linear operator on H, then we say that T is n-weakly hypercyclic if it has an orbit that is n-weakly dense in H and T is n-weakly supercyclic if T has a scaled orbit that is n-weakly dense in H. We will show that there are no 1-weakly hypercyclic matrices on F n for any n 1. Surprisingly, the most difficult case in this result is the case for 2 2 real matrices! One can easily check that an operator is 1-weakly supercyclic if and only if it is cyclic, so this is not a new concept. Thus, 1-weakly hypercyclic and 2-weakly supercyclic operators are the largest classes of Date: July 31, Mathematics Subject Classification. 47A16. Key words and phrases. n-weakly hypercyclic operator, n-weakly supercyclic operator. 1

2 2 NATHAN S. FELDMAN operators, or the weakest forms of dynamics beyond cyclicity, under consideration. It will be shown that there are 2-weakly supercyclic matrices on R n if and only if n is even, that there are no 2-weakly supercyclic matrices on C n, and that there are no 3-weakly supercyclic matrices on R n for n 3. Finally, we will show that there is a 2-weakly supercyclic onto isometry on a real infinite dimensional Hilbert space that is not 3-weakly supercyclic. We will also introduce the idea of n-weakly hypercyclic tuples of matrices and give some examples. We conclude with some open questions. 2. Preliminaries If H is a Hilbert space and if x 0 H, then a basis for the weak topology on H at the point x 0 is given as follows: For F = {f 1,..., f n } H and ɛ > 0, let N(x 0, F, ɛ) = N(x 0, f 1,..., f n, ɛ) = {x H : x x 0, f < ɛ for all f F}. If A is a set, then we will use A to denote the cardinality of A. Definition 2.1. Let n 1 be an integer, H a Hilbert space and E H. (1) E is n-weakly open if for every x 0 E there is an ɛ > 0 and a set F H with F n such that N(x 0, F, ɛ) E. (2) E H is n-weakly dense in H if E N for every non-empty n-weakly open set N H. (3) E is n-weakly closed if the complement of E is n-weakly open. (4) A point x 0 H is in the n-weak closure of a set E if for every n-weakly open set N containing x 0, we have N E. Notice that the sets of the form N(x 0, F, ɛ) are n-weakly open sets if the cardinality of F is at most n. In fact, they might properly be called the basic n-weakly open sets. But, be careful! The n-weakly open sets do not form a topology! They are not closed under finite intersections and n-weakly closed sets are not closed under finite unions. The following proposition gives some equivalent conditions for a set to be n- weakly dense which are useful. Condition (4) has a nice geometric appeal and both conditions (3) and (4) will be used often. Proposition 2.2. If H is a Hilbert space over F = R or C, E H, and n 1 is an integer, then the following are equivalent: (1) E is n-weakly dense in H. (2) F (E) is dense in F n for every onto continuous linear map F : H F n. x, f 1 (3). : x E is dense in Fn for every linearly independent set of x, f n vectors {f 1,..., f n } H. (4) The orthogonal projection of E onto M is dense in M for every subspace M of H with dimension n. It follows easily from the above proposition that if dim H n, then a set E H is n-weakly dense in H if and only if E is norm dense in H. One should think geometrically of a (basic) n-weakly open set N as an ɛ- neighborhood of an affine subspace L of codimension n. Thus a 1-weakly open set in R 2 is an ɛ-neighborhood of a line, hence a thin strip, or the region between

3 N-WEAKLY SUPERCYCLIC MATRICES 3 two parallel lines. A 2-weakly open set in R 2 is an ɛ-neighborhood of a point. A 1-weakly open set in R 3 is an ɛ-neighborhood of a plane, like a board or a plank, and a 2-weakly open set in R 3 is an ɛ-neighborhood of a line; like a long thin rod. With these images in mind, it easy to verify the following elementary examples. Any two non-parallel lines in R 2 form a 1-weakly dense set in R 2 that is not 2-weakly dense in R 2 and second, any three planes in R 3, no two of which are parallel (like the three coordinate planes), form a 2-weakly dense set in R 3 that is not 3-weakly dense in R n-weakly Hypercycylic/Supercyclic Operators A continuous linear operator T on a Hilbert space H over F = R or C is n-weakly hypercyclic if there is a vector x H such that Orb(x, T ) = {T n x} n=0 is n-weakly dense in H and T is n-weakly supercyclic if there is a vector x H such that the scaled orbit of x under T, F Orb(x, T ) = {ct n x : c F, n 0}, is n-weakly dense in H. Proposition 3.1. If T is n-weakly hypercyclic (respectively, n-weakly supercyclic), then the compression of T to a coinvariant subspace is also n-weakly hypercyclic (respectively, n-weakly supercyclic). The above proposition is equivalent to the following: If an operator T can be represented as a matrix with operator entries as A C T = 0 B and if T is n-weakly hypercyclic or n-weakly supercyclic, then the operator B also has the same property. This fact will be used repeatedly in what follows. The following useful result is proved in Shkarin [8, p. 62, Proposition 5.2] and in the book by Bayart and Matheron [3, p. 232]. The proofs are elementary consequences of a deep theorem due to Ball (see [1] and [2]). The statements do not use the term 1-weakly closed, but the proofs give the desired results. Theorem 3.2 (Ball s Theorem). If S = {x n } n=1 is a sequence of non-zero vectors in a Hilbert space H over F = R or C and if 1 n=1 x n <, then S is 1-weakly closed in H. The following proposition is elementary and routine except for part (3), which is routine but uses Ball s Theorem above. Proposition 3.3. If T is 1-weakly hypercyclic, then (1) T cannot have any (non-zero) bounded orbits. (2) T cannot have any eigenvectors. (3) Every component of the spectrum of T must intersect the unit circle. Shkarin [8] proved a weak angle criterion that can be used to show that an operator is not weakly supercyclic; also see [3, p. 240]. In fact, with only a small modification of his proof, we see that his weak angle criterion actually proves that the given vector is not a 2-weakly supercyclic vector. See Feldman [4] for the details.

4 4 NATHAN S. FELDMAN Theorem 3.4. (Weak Angle Criterion) Suppose that T is a bounded linear operator on a Hilbert space H and that x H. If there is a non-zero vector f H such that ( T n ) 2 x, f T n < x n=0 then x is not a 2-weakly supercyclic vector for T. The following result is an elementary consequence of the Weak Angle Criterion, see Feldman [4] for the details. Corollary 3.5 (Weak Ratio Criterion). If T is a bounded linear operator on a Hilbert space H and T has two invariant subspaces M and N that are complementary and such that for every x M and for every nonzero y N we have that ( T n ) 2 x T n < y n=0 then T is not 2-weakly supercyclic. Item (1) below follows from the Weak Ratio Criterion and item (2) follows easily from the definition of 2-weak supercyclicity. See Feldman [4] for the details. Proposition 3.6. Suppose that T is a bounded linear operator on a separable Banach space. (1) If T is 2-weakly supercyclic, then there is an r 0 such that every component of σ(t ) intersects the circle C = {z C : z = r}. (2) If T is 2-weakly supercyclic, then T cannot have two linearly independent eigenvectors. 4. Brief Review of Jordan Canonical Forms Recall that the k k Jordan block with eigenvalue λ, denoted by J k (λ), is a k k matrix with λ along the main diagonal and ones along the superdiagonal, and zeros elsewhere. Below is J 4 (λ). λ J 4 (λ) = 0 λ λ λ Powers of the matrix J k (λ) follow a simple pattern: λ n nλ n 1 n(n 1) 2 λ n 2 n(n 1)(n 2) J 4 (λ) n = 0 λ n nλ n 1 n(n 1) 2 λ n λ n nλ n λ n 3! λ n 3 The terms along the top row are simply 1 k! (λ n ) and the matrix is an uppertriangular Toeplitz matrix (constant along the diagonals). dλ k Every complex n n matrix T is similar to a Jordan matrix which is a direct sum of Jordan blocks J k (λ) of various sizes. However, if T is a real matrix, it s eigenvalues may be complex and thus it s Jordan form may have complex entries. d k

5 N-WEAKLY SUPERCYCLIC MATRICES 5 In that case the real Jordan form for T is useful; it uses the Jordan blocks J k (λ) when λ is real and some additional real blocks with complex eigenvalues. Let a b C(a, b) = b a then C(a, b) has eigenvalues a ± ib. If a + ib = re iθ = r cos(θ) + ir sin(θ) is the polar form of a + ib, then we also have a b r cos(θ) r sin(θ) cos(θ) sin(θ) C(a, b) = = = r = rr(θ) b a r sin(θ) r cos(θ) sin(θ) cos(θ) where cos(θ) sin(θ) R(θ) = sin(θ) cos(θ) is the rotation matrix that rotates by an angle of θ. A 2k 2k real Jordan block with complex eigenvalues, J k (r, θ), is the block upper-triangular matrix with k copies of rr(θ) = C(a, b) down the main diagonal and with 2 2 identity matrices on the super block-diagonal; which gives ones along the second-super-diagonal. Below are some examples: cos(θ) sin(θ) J 1 (r, θ) = r = rr(θ) sin(θ) cos(θ) r cos(θ) r sin(θ) 1 0 J 2 (r, θ) = r sin(θ) r cos(θ) 0 1 rr(θ) I r cos(θ) r sin(θ) = 0 rr(θ) r sin(θ) r cos(θ) r cos(θ) r sin(θ) 1 0 r sin(θ) r cos(θ) 0 1 J 3 (r, θ) = r cos(θ) r sin(θ) 1 0 r sin(θ) r cos(θ) 0 1 = r cos(θ) r sin(θ) r sin(θ) r cos(θ) rr(θ) I 0 = 0 rr(θ) I 0 0 rr(θ) Powers of these matrices follow the same pattern as for the J k (λ) blocks but with the observation that R(θ) n = R(nθ). Thus, we have J 3 (r, θ) n = rn R(nθ) nr n 1 n(n 1) R((n 1)θ) 2 r n 2 R((n 2)θ) 0 r n R(nθ) nr n 1 R((n 1)θ) 0 0 r n R(nθ) It is known that J m (r, θ) is similar to J m (λ) J m (λ) where λ = re iθ, see [6, p. 150]. Also, every real n n matrix T is similar to its real Jordan Form which is a direct sum of blocks J k (λ) where λ is a real eigenvalue for T and a direct sum of blocks of the form J k (r, θ) where [r cos(θ) ± ir sin(θ)] is a conjugate pair of complex eigenvalues for T. A Jordan matrix is any matrix that is a direct sum of Jordan blocks. For more information on the real Jordan form see [5, p. 359] or [6, p. 150].

6 6 NATHAN S. FELDMAN 5. n-weakly Hypercycylic Matrices Theorem 5.1. There are no 1-weakly hypercyclic operators on R n or C n. Proof. The complex case follows immediately from Proposition 3.3 since the adjoint of a 1-weakly hypercyclic operator cannot have any eigenvectors. In the real case, if T is 1-weakly hypercyclic on R n, then since T cannot have an eigenvector, T must have only complex eigenvalues, and thus the real Jordan form of T will be a direct sum of 2k 2k real Jordan blocks J k (r, θ) with complex eigenvalues. But then each of these direct summands must be 1-weakly hypercyclic; hence there is a k, r, and θ such that J k (r, θ) is 1-weakly hypercyclic. Then by compressing J k (r, θ) to the subspace spanned by the last two coordinates, and by applying Proposition 3.1, we see that J 1 (r, θ) must be 1-weakly hypercyclic. Note that J 1 (r, θ) is simply a multiple of a rotation matrix; J 1 (r, θ) = rr(θ). If r 1, then the orbits of J 1 (r, θ) are all bounded and hence, J 1 (r, θ) is not 1-weakly hypercyclic. If r > 1, then the orbits of J 1 (r, θ) increase exponentially fast. Thus by Ball s Theorem (Theorem 3.2), the orbits are 1-weakly closed, hence not 1-weakly dense in R 2. Thus J 1 (r, θ) is not 1-weakly hypercyclic. It follows that there are no 1-weakly hypercyclic operators on R n. It is interesting that the difficult case is the case of 2 2 real matrices; which is handled with Ball s deep Theorem. The fact that J 1 (r, θ) is not 1-weakly hypercyclic is equivalent to the fact that for the complex number z = re iθ, its sequence of powers, {z n } n=0, is not 1-weakly dense in C = R n-weakly Supercyclic Jordan Matrices Theorem 6.1 (n-weakly supercyclic Jordan matrices). (1) A Jordan block J n (λ) is not 2-weakly supercyclic on R n or C n when n 2. (2) A real Jordan block J k (r, θ) is not 2-weakly supercyclic on R 2k when k 2. (3) A real Jordan matrix of the form J k (r, θ) J 1 (λ) is not 2-weakly supercyclic on R 2k+1 when k 1. (4) J 1 (r 1, θ 1 ) J 1 (r 2, θ 2 ) is not 3-weakly supercyclic on R 4 when r 1 = r 2. (5) J 1 (r 1, θ 1 ) J 1 (r 2, θ 2 ) is not 2-weakly supercyclic on R 4 when r 1 r 2. Proof. (1) Let F be either R or C and assume that λ F. Suppose that J n (λ) is 2-weakly supercyclic, then by Proposition 3.1, the compression of J n (λ) to its last two [ coordinates ] is also 2-weakly supercyclic. However, this compression is λ 1 J 2 (λ) =. We can easily check that J 0 λ 2 (λ) is not 2-weakly supercyclic. For suppose that v = (z, w) is a 2-weakly supercyclic vector for J 2 (λ), then it follows that the scaled orbit of v is dense in F 2. Notice that { } λ E := Orb(v, J 2 (λ)) = c n z + nλ n 1 w λ n : c F, n 0 w which must be dense in F 2. It follows that w 0 and λ 0. However, we will show that the set E is not dense in F 2. Notice that E is simply a countable union of one-dimensional subspaces, so E F 2. We will show that any vector of the form [ a b] that is not in E and satisfies a 0 and b 0 does not belong to the closure of

7 N-WEAKLY SUPERCYCLIC MATRICES 7 the set E. To see this, suppose that there exist scalars {c k } in F and a sequence of positive integers n k such that [ λ n k z + n c k λ nk 1 w a k λ n k w b] where a 0 and b 0. Then we have b a = lim k c k λ n k w c k (λ n k z + nk λ n k 1 w) = lim k λw λz + n k w = 0. A contradiction since a, b 0. It follows that E is not dense in F 2. Thus J 4 (λ) is not 2-weakly supercyclic on F 4, and thus it J n (λ) is not 2-weakly supercyclic on F n. (2) Suppose, by way of contradiction, that there is a k 2, an r 0 and a θ R such that J k (r, θ) is 2-weakly supercyclic on R 2k. Then the compression of J k (r, θ) to last four coordinates (which is the 4 4 block matrix in the lower right hand corner of J k (r, θ) ) must also be 2-weakly supercyclic and that block is J 2 (r, θ). Then since r > 0 (because J 2 (r, θ) must have dense range) we can consider 1 r J 2(r, θ), which will have complex eigenvalues of modulus one and whose real Jordan form will be J 2 (1, θ). Thus, J 2 (1, θ), being similar to 1 r J 2(r, θ), will also be 2-weakly supercyclic on R 4. For convenience let T = J 2 (1, θ). The powers of T = J 2 (1, θ) are as follows: T n = J 2 (1, θ) n R(θ) n nr(θ) = n 1 R(θ) n = cos(nθ) sin(nθ) n cos((n 1)θ) n sin((n 1)θ) = sin(nθ) cos(nθ) n sin((n 1)θ) n cos((n 1)θ) cos(nθ) sin(nθ) sin(nθ) cos(nθ) Let M = {(x, y, 0, 0) : x, y R} and let N = {(0, 0, z, w) : z, w R}, then M and N are complementary invariant subspaces for T. Also, if v 1 M, then T n v 1 = v 1 and if v 2 = (0, 0, z, w) N, and v 3 = (z, w), then T n v 2 = nr(θ) n 1 v 3 R(θ) n = nr(θ) v n 1 v R(θ) n v 3 2 = nv v 3 2 = 3 = (n 2 + 1) v 3 2 = v 3 n = v 2 n n v 2. Thus, if v 1 M and v 2 N \ {0}, then ( ) T n 2 v 1 ( 2 n=0 T n v 2 = v1 n=0 n v 2 ) <. Thus it follows from the Weak Ratio Criterion (Corollary 3.5) that T is not 2-weakly supercyclic, and thus neither is J k (r, θ) for k 2. (3) Suppose by way of contradiction that for some k 1, there exist r > 0, θ R and λ R such that J k (r, θ) J 1 (λ) is 2-weakly supercyclic on R 2k+1. Then by compressing this matrix to the last three coordinates we see that r cos(θ) r sin(θ) 0 T = J 1 (r, θ) J 1 (λ) = r sin(θ) r cos(θ) λ

8 8 NATHAN S. FELDMAN must also be 2-weakly supercyclic on R 3, see Proposition 3.1. There are two cases to consider: r λ and r = λ. Notice that both r and λ must be non-zero for the above matrix to be 2-weakly supercyclic. Case 1: r λ. In this case we can easily apply the Weak Ratio Criterion (Corollary 3.5). Assume that 0 < r < λ, the other case when r > λ is similar. Then the powers of T have the following form: T n = rn cos(nθ) r n sin(nθ) 0 r n sin(nθ) r n cos(nθ) 0 r = n R(θ) n 0 0 λ n λ n Let M = {(x, y, 0) : x, y R} and let N = {(0, 0, z) : z R, then M and N are complementary invariant subspaces for T. Also, if v 1 M, then T n v 1 = r n v 1 and if v 2 N, then T n v 2 = λ n v 2. Thus, since r < λ, if v 1 M and v 2 N \ {0}, then n=0 supercyclic. ( T n v 1 T n v 2 ) 2 <. It then follows that T is not 2-weakly Case 2: r = λ. Since r 0, we may consider 1 r T, which is also 2-weakly supercyclic. Notice that S = 1 r T = J 1(1, θ) J 1 (±1) since λ/r = ±1 depending on whether λ > 0 or λ < 0. We ll consider the case λ < 0, the other case is similar. So we ll show that S = J 1 (1, θ) J 1 ( 1) is not 2-weakly supercyclic. This is a simple visual exercise! Indeed, any orbit of S does not have a dense projection onto the yz plane. To see this simply note that if v = (x, y, z) R 3, then the closure of the orbit is either equal to or is a finite subset of a union of two circles; namely the circles that are parallel to the xy-plane and pass through v and ˆv = (x, y, z). Thus, the scaled orbit is either a finite union of lines or a double cone and neither set has a dense projection onto the yz-plane. Thus S is not 2-weakly supercyclic. Alternatively, one may give a more analytic arugment by showing that if v = (x, y, z) is a 2-weakly supercyclic vector for T, then { c [ T n v, e 2 T n v, e 3 ] } : c R, n 0 = { c x sin(nθ) + y cos(nθ) λ n z } : c R, n 0 is not dense in R 2 where e 2 = (0, 1, 0) and e 3 = (0, 0, 1). We leave that to the reader. (4) Here we will show that T = J 1 (r 1, θ 1 ) J 1 (r 2, θ 2 ) is not 3-weakly supercyclic on R 4 when r 1 = r 2. Since r 1 = r 2, we may divide out the common value and assume that r 1 = r 2 = 1. Thus it suffices to show that if T = J 1 (1, θ 1 ) J 1 (1, θ 2 ), then T is not 3-weakly supercyclic. Suppose T is 3-weakly supercyclic. Notice that T n has the following form: cos(nθ 1 ) sin(nθ 1 ) T n = sin(nθ 1 ) cos(nθ 1 ) cos(nθ 2 ) sin(nθ 2 ) sin(nθ 2 ) cos(nθ 2 ) = R(θ1 ) n R(θ 2 ) n Suppose that v = (x, y, z, w) is a 3-weakly supercyclic vector for T. Let e 1 = (1, 0, 0, 0), e 2 = (0, 1, 0, 0), and e 3 = (0, 0, 1, 0). We claim that 0 0 is not in the 1

9 N-WEAKLY SUPERCYCLIC MATRICES 9 closure of ct n v, e 1 ct n v, e 2 : c R, n 0 ct n. v, e 3 Suppose that there are constants {c k } such that c kt n k v, e 1 c k T n k v, e as k. c k T n k v, e 3 1 Then since the first two coordinates converge to zero, we have c k R(θ 1 ) n k v 1 0 as k where v 1 = (x, y). Since R(θ 1 ) is an isometry, we have that c k = c k R(θ 1 ) n k v 1 0 as k. But since c k 0, one easily sees that c k T n k v, e 3 0 as k, contradicting the assumption above that it converges to 1. Thus, T is not 3-weakly supercyclic; and thus J 1 (r 1, θ 1 ) J 1 (r 2, θ 2 ) is not 3-weakly supercyclic when r 1 = r 2. (5) Here we will show that T = J 1 (r 1, θ 1 ) J 1 (r 2, θ 2 ) is not 2-weakly supercyclic on R 4 when r 1 r 2. Without loss of generality, assume that 0 < r 1 < r 2. Notice that T n will have the following form: r1 n cos(nθ 1 ) r1 n sin(nθ 1 ) T n = r1 n sin(nθ 1 ) r1 n cos(nθ 1 ) r2 n cos(nθ 2 ) r2 n sin(nθ 2 ) r2 n sin(nθ 2 ) r2 n cos(nθ 2 ) Let M = {(x, y, 0, 0) : x, y R} and N = {(0, 0, z, w) : z, w R}, then M and N are complementary invariant subspaces for T. Also, T n v 1 = r1 n v 1 for v 1 M and T n v 2 = r2 n v 2 for v 2 N. Thus, if v 1 M and v 2 N \ {0}, then ( r n 1 v1 r n 2 v2 ) 2 < since r1 < r 2. Thus the Weak Ratio ( ) T n 2 v 1 n=0 T n v 2 = n=0 Criterion (Corollary 3.5) implies that T is not 2-weakly supercyclic. 7. n-weakly Supercyclic Matrices It s well known that a 2 2 irrational rotation matrix on R 2 is supercyclic and hence 2-weakly supercyclic. It s also known that there are no supercyclic matrices on C n when n 2 and no supercyclic matrices on R n when n 3. We will refine these results below and show the surprising fact that there are 2-weakly supercyclic matrices on R n if and only if n is even. It is an easy exercise to see that an operator is 1-weakly supercyclic if and only if it is cyclic. And cyclic matrices are well understood and exist in every dimension. Theorem 7.1 (n-weakly supercyclic matrices). (1) There are no 2-weakly supercyclic operators on C n for n 2. (2) There are no 3-weakly supercyclic operators on R n for n 3. Proof. (1) By way of contradiction, suppose that there exists an operator T on C n, n 2, that is 2-weakly supercyclic. Then by Proposition 3.6, T cannot have two independent eigenvectors. Thus the Jordan form of T consists of a single Jordan block, J n (λ). Thus J n (λ) is 2-weakly supercyclic on C n with n 2, but this contradicts item (1) of Theorem 6.1. (2) Suppose, by way of contradiction, that T is a 3-weakly supercyclic operator on R n where n 3. Then by Proposition 3.6, T cannot have two independent

10 10 NATHAN S. FELDMAN eigenvectors. So T can have at most one real eigenvalue. Thus there are two cases to consider: either T has one real eigenvalue or T has no real eigenvalues. Notice that any eigenvalues must be non-zero, since T must have dense range. Case 1: T has one real eigenvalue, call it λ. It follows that the Jordan form of T has a k k Jordan block, J k (λ) with a real eigenvalue λ, where 1 k n. There are two cases to consider here. Either k 2 or k = 1. Case 1a: k 2. In this case, the k k Jordan block J k (λ) must be 3-weakly supercyclic on R k, but this contradicts item (1) of Theorem 6.1 which says that J k (λ) is not even 2-weakly supercyclic. Case 1b: k = 1. In this case, the 1 1 block J 1 (λ) is a block in the real Jordan form for T and all other blocks in the real Jordan form of T must be of the form J m (r, θ) for some m 1 and some values of r and θ. Also since n 3, at least one such block exists. Since T is 3-weakly supercyclic, then the direct sum of any collection of Jordan blocks for T will also be 3-weakly supercyclic. In particular, if J m (r, θ) is a Jordan block for T, then the (2m + 1) (2m + 1) matrix J = J m (r, θ) J 1 (λ) must be 3-weakly supercyclic on R 2m+1. However this contradicts item (3) of Theorem 6.1, which in fact says that such matrices cannot be 2-weakly supercyclic. Case 2: T has no real eigenvalues. Consider the real Jordan form of T, since T has no real eigenvalues, then every real Jordan block for T has the form J m (r, θ) for some m 1. By item (2) of Theorem 6.1, there can be no such blocks with m 2 (i.e. size at least 4 4). Hence each block is a 2 2 block of the form J 1 (r, θ), and the real Jordan form for T is a direct sum of such 2 2 blocks, hence n must be even. Since n is even and since we are assuming that n 3, then that means that n 4, so that there are at least two 2 2 Jordan blocks in the real Jordan form. It follows that the direct sum of any two of the 2 2 Jordan blocks from the Jordan form of T must be 3-weakly supercylic on R 4. But this contradicts item (4) or (5) of Theorem 6.1. It follows that there are no 3-weakly supercyclic matrices on R n when n Weakly Supercyclic Matrices on R n We now move toward proving that there are 2-weakly supercyclic matrices on R n when n is even and give a partial characterization of such matrices. By a ray from the origin we mean a set of the form R = {tu : t > 0} where u is a non-zero vector in R 2. We will also say that a set E R 2 surrounds the origin if for every ray R from the origin we have R E. Clearly if a set E surrounds the origin, then the set of multiples of E, R E, is equal to R 2. In fact, positive multiples will suffice. We define the sum of two subsets X and Y of a vector space in the natural way as X + Y = {x + y : x X, y Y }, finite sums of sets are defined similarly. Lemma 8.1. If K j, 1 j n are sets in R 2 such that for each 1 j n, the set K j surrounds the origin, then the sum K 1 + K K n also surrounds the origin. Proof. If u is a non-zero vector in R 2 and R is the ray from the origin determined by u, then since K j surrounds the origin, R K j. Say, v j R K j. Then v j = t j u for some t j > 0. Thus, (v 1 + v v n ) (K 1 + K K n ) and

11 N-WEAKLY SUPERCYCLIC MATRICES 11 (v 1 + v v n ) = (t 1 + t t n )u R, thus (K 1 + K K n ) R. It follows that (K 1 + K K n ) surrounds the origin. Let T = {(x, y) R 2 : x 2 + y 2 = 1} and let T n = T T T (n times). So, elements of T n are n-tuples of vectors of size two, e.g. v = ((x, y), (z, w)) T 2. There is a natural identification of T n with a subset ˆT n of R 2n as follows: v = ((x 1, x 2 ), (x 3, x 4 ),..., (x 2n 1, x 2n ) (x 1, x 2, x 3, x 4,..., x 2n 1, x 2n ) ˆT n R 2n. Proposition 8.2. The set R + ˆT n is 2-weakly dense in R 2n. Proof. In order to show that R + ˆT n = {c v : c R +, v ˆT n } is 2-weakly dense in R 2n, by Proposition 2.2, it suffices to show that { } x, f c : c R +, x x, g ˆT n is dense in R 2 for any two linearly independent (hence nonzero) vectors f, g R 2n. In fact, we will show that it is equal to R 2. Write f = (f 1,..., f 2n ) and g = (g 1,..., g 2n ) and consider the matrix f1 f A = 2 f 3 f 4 f 2n 1 f 2n. g 1 g 2 g 3 g 4 g 2n 1 g 2n x, f Notice that Ax = for all x R x, g 2n. So we must show that R + A(ˆT n ) is dense in R 2. Partition A into 2 2 blocks as follows: f1 f A = 2 f 3 f 4 f 2n 1 f 2n = [A g 1 g 2 g 3 g 4 g 2n 1 g 1 A 2 A n ]. 2n then we have the following: Thus, A(ˆT n ) = A 1 (T) + A 2 (T) + + A n (T). R + A(ˆT n ) = R + [A 1 (T) + A 2 (T) + + A n (T)]. Since each A i is a 2 2 matrix, then depending on the rank of A i, A i (T) is either {0}, a non-trivial line segment in R 2 centered at 0, or an ellipse centered at 0, depending on whether rank(a i ) equals zero, one, or two respectively. There are two cases to consider: either at least one of the matrices A i has rank two or they all have rank at most one. Case 1: rank(a i ) = 2 for at least one i {1,..., n}. Let J = {i {1,..., n} : rank(a i ) = 2}. Then J and for each i J, A i (T) is an ellipse in the plane centered at the origin, and thus surrounds the origin. Hence, by Lemma 8.1, the sum i J A i(t) is a set that surrounds the origin. Furthermore, for each i {1,..., n} \ J we have that 0 A i (T). Thus, n A i (T) A i (T) = A(ˆT n ). i J i=1 It follows that A(ˆT n ) surrounds the origin and hence R + A(ˆT n ) is equal to R 2, as desired. Case 2: rank(a i ) 1 for all i {1,..., n}.

12 12 NATHAN S. FELDMAN This time define J = {i {1,..., n} : rank(a i ) = 1}. Since f and g are nonzero, then J. We claim that at least two of the matrices {A i } will have rank one. If not, then J = {i 0 } for some i 0 {1,..., n}. Thus, rank(a i ) = 0 for all i i 0. Hence A i is the zero matrix when i i 0 and A i0 has rank one. So the two rows of A i0 are multiples of each other and since A i = 0 when i i 0, then that implies that f and g are multiples of each other, contradicting the fact that they are independent. So, J must have at least two elements. For each i J, the rows of A i are multiples of each other. Let k i be the scalar such that the first row in A i is equal to k i times the second row (row 1 = k i row 2 ). Since the vectors f and g are independent, reasoning as above, there must be i, j J such that k i k j. Let 1 ki B =. 1 k j Notice that B is invertible since k i k j and also we have that 0 0 a b BA i = and BA c d j = 0 0 for some scalars a, b, c, d where (c, d) (0, 0) and (a, b) (0, 0), thus a 2 + b 2 > 0 and c 2 + d 2 > 0. Notice that { } 0 BA i (T) = : y (c 2 + d 2 ) y and Thus, BA j (T) = {[ x BA i (T) + BA j (T) = y] { } x : x (a 2 + b 2 ). 0 } : x (a 2 + b 2 ) and y (c 2 + d 2 ). It follows that BA i (T)+BA j (T) is a rectangle in R 2 that contains zero in its interior. Since BA = [BA 1 BA 2 BA n ] it follows that BA(ˆT n ) = n k=1 BA k(t). Also, since we are in Case 2, BA k (T) contains the zero vector for every k, so it follows that n BA i (T) + BA j (T) BA k (T) = BA(ˆT n ). k=1 Thus BA(ˆT n ) contains a rectangle that contains the origin in its interior, hence R + BA(ˆT n ) = R 2. Thus, B(R + A(ˆT n )) = R 2, however since B is invertible it follows that R + A(ˆT n ) = R 2, as desired. Corollary 8.3. If {c k } n k=1 are non-zero real numbers, then all positive scalar multiples of the set c 1 T c 2 T c n T, considered naturally as a subset of R 2n, is 2-weakly dense in R 2n. Proof. By Proposition 8.2 we have that R + ˆT n is 2-weakly dense in R 2n. If D is the diagonal matrix whose diagonal entries are as follows: (c 1, c 1, c 2, c 2, c 3, c 3,..., c n, c n ), then D is a 2n 2n diagonal matrix which is invertible since c k 0 for all 1 k n. Thus D(R + ˆT n ) = R + D(ˆT n ) must be 2-weakly dense in R 2n. Since D(ˆT n ) is the subset of R 2n that is naturally identified with c 1 T c 2 T c n T, the result follows.

13 N-WEAKLY SUPERCYCLIC MATRICES 13 Theorem 8.4. If T = p k=1 J 1(r, θ k ), r > 0, on R 2p and if the set {π, θ 1, θ 2,..., θ p } is linearly independent over the the field Q of rational numbers, then T is 2-weakly supercyclic on R 2p. Proof. By taking a multiple of T we may assume that r = 1, in which case we have that cos(θ 1 ) sin(θ 1 ) sin(θ 1 ) cos(θ 1 ) T =... cos(θp) sin(θp) sin(θ p ) cos(θ p ) and if we let v = (1, 0, 1, 0, 1, 0,..., 1, 0), then we claim that v is a 2-weakly supercyclic vector for T. Notice that cos(nθ 1 ) sin(nθ 1 ) T n v =. cos(nθ p ) sin(nθ p ) Since the set {π, θ 1, θ 2,..., θ n } is linearly independent over Q, Kroneker s Theorem [7] implies that the closure of the orbit of v, Orb(v, T ), is equal to ˆT p. Since Proposition 8.2 implies that R ˆT p is 2-weakly dense in R 2p, it follows that R Orb(v, T ) is 2-weakly dense in R 2p, thus T is 2-weakly supercyclic. Also, any multiple of the above matrices are 2-weakly supercyclic. We can also easily classify the 2-weakly supercyclic vectors for T = J 1 (r, θ 1 ) J 1 (r, θ p ). Proposition 8.5. If T = p k=1 J 1(r, θ k ) is 2-weakly supercyclic on R 2p, then a vector w = (x 1,..., x 2p ) is a 2-weakly supercyclic vector for T if and only if (x 2k 1, x 2k ) (0, 0) for all 1 k p. Proof. It is easy to see that the condition is necessary. For the converse, simply apply the following observation. If v is a 2-weakly supercyclic vector for T and if S = c 1 S 1 c 2 S 2 c p S p where each S k is a rotation matrix and c k > 0, then S commutes with T and is invertible, so Sv is also a 2-weakly supercyclic vector for T. Theorem 8.6. There are 2-weakly supercyclic operators on R n if and only if n is even. Proof. If n is even, then the matrices from Theorem 8.4 are 2-weakly supercyclic. For the converse, suppose that T is a 2-weakly supercyclic operator on R n, then we must show that n is even. We may assume that n 3. In fact, we will show that T cannot have a real eigenvalue, and thus it follows that n must be even. Suppose, by way of contradiction that T does have have a real eigenvalue, call it λ. By Proposition 3.6, T cannot have two independent eigenvectors, so λ is the only real eigenvalue for T and λ will have a one-dimensional eigenspace. With this information, let s determine the real Jordan form of T. Since the Jordan block J k (λ) is not 2-weakly supercyclic when k 2 (by Theorem 6.1 item (1)), then the real Jordan form for T can only have λ appearing in a single 1 1 block J 1 (λ). Since n 3, then there is at least one other block in the real Jordan form of T

14 14 NATHAN S. FELDMAN and since T has no other real eigenvalues, it must be a block of the form J m (r, θ) where m 1. Thus J m (r, θ) J 1 (λ) is a part of the Jordan form for T and thus it must be 2-weakly supercyclic; but this contradicts item (3) of Theorem 6.1. Thus it must be that T has no real eigenvalues; and so n must be even. The following result almost characterizes which matrices on R 2n are 2-weakly supercyclic. Exactly how much independence among angles is necessary is all that remains open. The conditions stated are given when considering the matrix as a matrix acting on C 2n. Theorem 8.7. An operator T on R 2n is 2-weakly supercyclic if (1) T is diagonalizable over C. (2) All the eigenvalues of T are imaginary (a+bi with b 0) and all eigenvalues have the same absolute value. (3) The set {π, θ 1, θ 2,..., θ n } is linearly independent over Q, where {θ j } n j=1 are the arguments of the eigenvalues of T with positive imaginary part. Conversely, if T is an operator on R 2n that is 2-weakly supercyclic, then the following hold: (1 ) T is diagonalizable over C. (2 ) All the eigenvalues of T are imaginary (a+bi with b 0) and all eigenvalues have the same absolute value. (3 ) No two numbers from the set {π, θ 1, θ 2,..., θ n } are rational multiples of each other, where {θ j } n j=1 are the arguments of the eigenvalues of T with positive imaginary part. Notice that conditions (3) and (3 ) are not the same. It is not known if (3) is necessary or if (3 ) is sufficient or something in between. Proof. If conditions (1), (2), and (3) hold, then the real Jordan form of T is equal to n k=1 J 1(r, θ k ) and it then follows from the proof of Theorem 8.6 that T is 2-weakly supercyclic. Conversely, suppose that T is a 2-weakly supercyclic operator on R 2n, then by the proof of Theorem 8.6, T has no real eigenvalues, hence they are all imaginary. If T is not diagonalizable as a matrix over C, then the real Jordan form must have a real Jordan block of the form J m (r, θ) for some m 2. Thus the block J m (r, θ) would be 2-weakly supercyclic, but this contradicts item (2) of Theorem 6.1. Hence T is diagonalizable over C. Since T is diagonalizable over C and only has imaginary eigenvalues, the real Jordan form of T is a direct sum of Jordan blocks of the form J 1 (r k, θ k ) for 1 k n. It follows from Theorem 6.1 part (5) that r k = r j for all k, j. From which we see that all the eigenvalues of T must have the same absolute value each θ k must be an irrational multiple of π since each block J 1 (r k, θ k ) must be 2-weakly supercyclic on R 2 (which means supercylic on R 2 ). Also, if θ k = αθ j where α Q, then by Proposition 8.5, v = (1, 0, 1, 0,..., 1, 0) is a 2-weakly supercyclic vector for T and we since we may assume that r k = 1 for all k, it{ follows [ that the ] Orb(v, T ) projected } onto two appropriate coordinates cos(nθj ) gives that c : n 0, c R must be dense in R cos(nαθ j ) 2. However, it is not dense and this can be seen by examining the set {(z, z α ) : z T} and seeing that it is not dense in T 2 (here z α denotes all (finitely many) possible values of z α ).

15 N-WEAKLY SUPERCYCLIC MATRICES Weakly Supercyclic Operators in Infinite Dimensions In this section we will show that there is a real onto linear isometry on l 2 R (N) that is 2-weakly supercyclic but not 3-weakly supercyclic. By l 2 R (N) we mean the real Hilbert space of all square summable real sequences indexed by the positive integers. We will say that an infinite set of real numbers is linearly independent over Q if every finite subset of it is linearly independent over Q. Theorem 9.1. If {π, θ 1, θ 2,..., θ n,... } is a infinite sequence that is linearly independent over Q, then T = J 1 (1, θ k ) is 2-weakly supercyclic on l 2 R (N). k=1 Proof. Let v = (1, 0, 1/2, 0, 1/3, 0, 1/4, 0,..., 1/n, 0,...). Notice that v l 2 R (N). Since the set {π, θ 1, θ 2,..., θ n } is linearly independent over Q for every n 1, it follows from Theorem 8.4 that T n = n k=1 J 1(1, θ k ) is 2-weakly supercyclic on R 2n for every n. Also, by Proposition 8.5, we see that v n = (1, 0, 1/2, 0,..., 1/n, 0) is a 2-weakly supercyclic vector for T n. This fact, together with a simple approximation argument, shows that T is 2-weakly supercyclic on l 2 R (N). Notice that a simple countability argument shows that there exist angles {θ k } k=1 such that the set {π} {θ k } k=1 is linearly independent over Q. More specifically, since π is a transcendental number, the sequence {π k } k=1 is linearly independent over Q. 10. Tuples of Matrices If T = (T 1, T 2,..., T k ) is a commuting tuple of matrices acting on F d where F = R or C, then we say that T is n-weakly hypercyclic if there is a vector x F d such that Orb(v, T ) = {T n1 1 T n2 2 T n k k x : n j 0} is dense in F d. Proposition If {π, θ 1, θ 2,..., θ k } is linearly independent over Q, I denotes the identity matrix, and k T 1 = J 1 (1, θ j ) j=1 then the pair T = (T 1, 2I) is 1-weakly hypercyclic on R 2k and the triple T = (T 1, 2I, 1 3 I) is 2-weakly hypercyclic on R2k. This result follows easily from the results of this paper. The proof uses Proposition 8.2 and the fact that {2 n /3 k : n, k 0} is dense in R +. One may also let T 1 = J 1 (1/5) k j=1 J 1(1, θ j ), then (T 1, 2I) is 1-weakly hypercyclic on R 2k Questions (1) If T = n k=1 J 1(r, θ k ) is 2-weakly supercyclic on R 2n, then must the set {π, θ 1, θ 2,..., θ n } be linearly independent over Q? (2) When 1 n < d, what is the smallest value of k such that there is a commuting k-tuple of matrices on R d (or C d ) that is n-weakly hypercyclic? (3) If 1 n < d, then is there a commuting tuple of matrices on F d that is n-weakly hypercyclic but not (n + 1)-weakly hypercyclic?

16 16 NATHAN S. FELDMAN References [1] K. Ball, The plank problem for symmetric bodies, Invent. Math. 104 (1991), [2] K. Ball, The complex plank problem, Bull. London Math. Soc. 33 (2001), [3] F. Bayart & E. Matheron, Dynamics of Linear Operators, Cambridge University Press, [4] N. S. Feldman, n-weakly Hypercyclic & n-weakly Supercyclic Operators, preprint. [5] I. Gohberg, P. Lancaster, & L. Rodman, Invariant Subspaces of Matrices with Applications, Canadian Mathematical Society, John Wiley & Sons, [6] R.A. Horn & C.R. Johnson, Matrix Analysis, Cambridge University Press, [7] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, [8] S. Shkarin, Non-sequential weak supercyclicity and hypercyclicity, J. Funct. Anal., 242, (2007), Dept. of Mathematics, Washington and Lee University, Lexington VA address: