This paper explores the trade-off between maximizing the secondary user's transmit throughput and minimizing interference to primary receivers in cognitive radio (CR) networks. The authors characterize the secondary user's channel capacity under transmit-power and interference-power constraints using an information-theoretic approach. They exploit multi-antennas at the secondary transmitter to balance spatial multiplexing and interference avoidance. Convex optimization techniques are used to design algorithms for optimal transmit spatial spectrum, and suboptimal solutions are also presented for implementation ease. The algorithms are extended to multi-channel transmission, allowing the secondary user to adapt resources in space, time, and frequency domains. Simulation results show that substantial capacity gains are achievable even under stringent interference-power constraints, and an "interference diversity" effect is observed when the number of primary receivers exceeds the number of secondary transmit antennas.This paper explores the trade-off between maximizing the secondary user's transmit throughput and minimizing interference to primary receivers in cognitive radio (CR) networks. The authors characterize the secondary user's channel capacity under transmit-power and interference-power constraints using an information-theoretic approach. They exploit multi-antennas at the secondary transmitter to balance spatial multiplexing and interference avoidance. Convex optimization techniques are used to design algorithms for optimal transmit spatial spectrum, and suboptimal solutions are also presented for implementation ease. The algorithms are extended to multi-channel transmission, allowing the secondary user to adapt resources in space, time, and frequency domains. Simulation results show that substantial capacity gains are achievable even under stringent interference-power constraints, and an "interference diversity" effect is observed when the number of primary receivers exceeds the number of secondary transmit antennas.