Thank you for your question.
I assume that the arrow that you mean is the Enter key. Pressing it, however would not execute the ActionButtonCommand, because for some scenarios (for example multiple row text box) it might not be the desired behavior. You can easily implement this functionality by providing an event handler for the KeyDown or KeyUp events of RadTextBox and execute the associated command manually:
sender, KeyEventArgs e)
(e.Key == Key.Enter)
ICommand command = textBox.ActionButtonCommand
I hope this information helps. Let me know if I can assist you further.
the Telerik team