For a given function ϕ ∈ H ∞ with ϕ (z) < 1 (z ∈ D), we associate some special operators subspace and study some properties of these operators including behavior of their Berezin symbols. It turns that such boundary behavior is closely related to the Blaschke condition of sequences in the unit disk D of the complex plane. In terms of Berezin symbols the trace of some nuclear truncated Toeplitz operator is also calculated,