On rank-one asymmetric truncated Toeplitz operators on finite-dimensional model spaces

Characterizations of Asymmetric Truncated Toeplitz and Hankel Operators

“…Gu showed in [7] that a bounded linear operator B on K α is a truncated Hankel operator if and only if B S α − S * α B is a special kind of operator of rank at most two. Similar characterizations for the operators from T (α, β) were proved for the case when β divides α (that is, α/β is an inner function) [1], and for the case when α and β are finite Blaschke products (in other words, when K α and K β are finitely dimensional) [10].…”

Asymmetric Truncated Toeplitz Operators of Rank One

“…For example, he proved that a bounded linear operator A on K α is a truncated Toeplitz operator if and only if A − S α AS * α is a rank-two operator of a special kind, where S α = A α z is the so-called compressed shift. Similar characterizations of asymmetric truncated Toeplitz operators were given in [2,3] and [9] for some particular cases. Here, we use the following two characterizations of asymmetric truncated Toeplitz operators.…”

“…Nevertheless, it was also showed in [9] that the answer is positive for all other finite-dimensional cases (that is, whenever α and β are two finite Blaschke products): In this paper, we resolve the issue in the general case. In particular, we prove that if both K α and K β have dimension larger than one (and not necessarily finite), then the only rank-one operators in T (α, β) are the non-zero scalar multiples of k …”

“…This issue was addressed in [9]. The author in [9] proved that if K α is one dimensional and K β has a finite dimension larger than two (or K β is one dimensional and dim K α > 2), then the answer is negative: there exists a rank-one operator in T (α, β) that is neither a scalar multiple of k…”

“…So, here we take the approach from [9], that is, we use the following lemma (compare with [9,Lem. 4.4]; see also [10, pp.…”