On Turnbull identity for skew-symmetric matrices

In the last six lines of Turnbull's 1948 paper, he left an enigmatic statem ent on a Capelli-type identity for skew-symmetric matrix spaces. In the pre sent paper, on Turnbull's suggestion, we show that certain Capelli-type ide ntities hold for this case. Our formulae connect explicitly the central ele ments in U(gl(n)) to the invariant differential operators, both of which ar e expressed with permanent. This also clarifies the meaning of Turnbull's s tatement from the Lie-theoretic point of view.